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and a reflection across the x-axis. Another way to identify the stretch is to notice that the yvalues in the table are –5 times the corresponding yvalues for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}. 10-1 Square Root Functions Graph each function. Compare to the parent graph. State the domain and range. 3. 1. SOLUTION: x 0 0.5 y 0 ≈ 2.1 1 3 2 ≈ 4.2 3 ≈ 5.2 SOLUTION: x 0 0.5 y 0 ≈ 0.2 4 6 2 ≈ 0.5 3 ≈ 0.6 4 ≈ 0.7 The parent function is multiplied by a value less than 1 and greater than 0, so the the graph is a vertical compression of . Another way to identify the compression is to notice that the y-values The parent function is multiplied by a value greater than 1, so the graph is a vertical stretch of . Another way to identify the stretch is to notice that the y-values in the table are 3 times the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}. 2. 1 ≈ 0.3 in the table are times the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}. 4. SOLUTION: x 0 0.5 y 0 ≈ –3.5 1 –5 2 ≈ –7.1 3 ≈ –8.7 4 –10 SOLUTION: x 0 0.5 y 0 ≈ –0.4 The parent function is multiplied by a value less than 1, so the graph is a vertical stretch of and a reflection across the x-axis. Another way to identify the stretch is to notice that the yvalues in the table are –5 times the corresponding yvalues for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}. 3 ≈ –0.9 2 ≈ 0.5 3 ≈ 0.6 4 ≈ 0.7 4 –1 y-values in the table are times the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}. 5. eSolutions Manual - Powered by Cognero 1 ≈ 0.3 2 ≈ –0.7 The parent function is multiplied by a value less than 1, so the graph is a vertical compression of and a reflection across the x-axis. Another way to identify the compression is to notice that the 3. SOLUTION: x 0 0.5 y 0 ≈ 0.2 1 –0.5 Page 1 SOLUTION: x 0 0.5 1 2 3 4 units from the parent graph . Another way to identify the translation is to note that the y-values in the table are 2 less than the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ –2}. way to identify the compression is to notice that the y-values in the table are times the corresponding 10-1y-values Square for Root theFunctions parent function. The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}. 5. 7. SOLUTION: x 0 0.5 y 3 ≈ 3.7 1 4 2 ≈ 4.4 3 ≈ 4.7 SOLUTION: x –2 y 0 4 5 –1 1 0 ≈ 1.4 1 ≈ 1.7 2 2 The value 3 is being added to the parent function , so the graph is translated up 3 units from The value 2 is being added to the square root of the parent function , so the graph is translated 2 the parent graph . Another way to identify the translation is to note that the y-values in the table are 3 greater than the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 3}. units left from the parent graph . Another way to identify the translation is to note that the xvalues in the table are 2 less than the corresponding x-values for the parent function. The domain is {x|x ≥ –2}, and the range is {y|y ≥ 0}. 6. 8. SOLUTION: x 0 0.5 y –2 ≈ – 1.3 1 –1 2 ≈ – 0.6 3 ≈ – 0.3 4 0 The value 2 is being subtracted from the parent function , so the graph is translated down 2 units from the parent graph . Another way to identify the translation is to note that the y-values in the table are 2 less than the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ –2}. 4 1 5 ≈ 1.4 6 ≈ 1.7 7 2 The value 3 is being subtracted from the square root of the parent function , so the graph is translated 3 units right from the parent graph . Another way to identify the translation is to note that the x-values in the table are 3 more than the corresponding x-values for the parent function. The domain is {x|x ≥ 3}, and the range is {y|y ≥ 0}. 9. FREE FALL The time t, in seconds, that it takes an object to fall a distance d, in feet, is given by the 7. SOLUTION: eSolutions - Powered–1 by Cognero0 x Manual–2 y 0 1 ≈ 1.4 SOLUTION: x 3 y 0 function 1 ≈ 1.7 2 2 (assuming zero air resistance). Page 2 Graph the function, and state the domain and range. SOLUTION: translated 3 units right from the parent graph . Another way to identify the translation is to note that the x-values in the table are 3 more than the 10-1corresponding Square Rootx-values Functions for the parent function. The domain is {x|x ≥ 3}, and the range is {y|y ≥ 0}. The domain is {d | d ≥ 0} and the range is {t | t ≥ 0}. 9. FREE FALL The time t, in seconds, that it takes an object to fall a distance d, in feet, is given by the Graph each function, and compare to the parent graph. State the domain and range. function (assuming zero air resistance). 10. Graph the function, and state the domain and range. SOLUTION: x 0 0.5 y 2 ≈ 2.4 SOLUTION: Let d = 0, 1, 4, 9, and 16 feet. Make a table to find accompanying values for t. d t 0 0 1 2.5 2 ≈ 2.7 3 ≈ 2.9 4 3 1 4 9 16 The parent function is multiplied by a value less than 1 and is added to the value 2, so the graph is a vertical compression of followed by a translation 2 units up. The domain is {x|x ≥ 0}, and the range is {y|y ≥ 2}. 1 Plot each point (d, t) and connect them with a smooth curve. 11. SOLUTION: x 0 0.5 y –1 ≈ – 0.8 1 –1.25 2 ≈ – 1.35 3 ≈ – 1.4 4 –1.5 The domain is {d | d ≥ 0} and the range is {t | t ≥ 0}. Graph each function, and compare to the parent graph. State the domain and range. The graph is the result of a vertical compression of the graph of y = followed by a reflection across the x-axis, and then a translation 1 unit down. The domain is {x|x ≥ 0}, and the range is {y|y ≤ –1}. 10. SOLUTION: x 0 0.5 y 2 ≈ 2.4 1 2.5 eSolutions Manual - Powered by Cognero 2 ≈ 2.7 3 ≈ 2.9 4 3 12. SOLUTION: x –1 y 0 0 –2 1 ≈ –2.8 2 ≈ –3.5 3 Page 3 –4 The graph is the result of a vertical compression of This graph is the result of a vertical stretch of the the graph of y = followed by a reflection across 10-1the Square Functions x-axis,Root and then a translation 1 unit down. The domain is {x|x ≥ 0}, and the range is {y|y ≤ –1}. graph of y = followed by a translation 2 units right. The domain is {x|x ≥ 2}, and the range is {y|y ≥ 0}. Graph each function. Compare to the parent graph. State the domain and range. 12. SOLUTION: x –1 y 0 14. 0 –2 1 ≈ –2.8 2 ≈ –3.5 3 –4 SOLUTION: x 0 0.5 y 0 ≈ 3.5 This graph is the result of a vertical stretch of the graph of followed by a reflection across the x-axis, and then a translation 1 unit left. The domain is {x|x ≥ –1}, and the range is {y|y ≤ 0}. 3 3 4 ≈ 4.2 5 ≈ 5.2 2 ≈ 7.1 3 ≈ 8.7 4 10 The parent function is multiplied by a value greater than 1, so the graph is a vertical stretch of . Another way to identify the stretch is to notice that the y-values in the table are 5 times the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}. 13. SOLUTION: x 2 y 0 1 5 6 6 15. SOLUTION: x 0 0.5 y 0 ≈ 0.35 1 0.5 2 ≈ 0.7 3 ≈ 0.9 4 1 This graph is the result of a vertical stretch of the graph of y = followed by a translation 2 units right. The domain is {x|x ≥ 2}, and the range is {y|y ≥ 0}. Graph each function. Compare to the parent graph. State the domain and range. The parent function is multiplied by a value less than 1 and greater than 0, so the the graph is a vertical compression of . Another way to identify the compression is to notice that the y-values 14. SOLUTION: x 0 0.5 y 0 ≈ 3.5 1 5 eSolutions Manual - Powered by Cognero 2 ≈ 7.1 3 ≈ 8.7 4 10 in the table are times the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}. 16. Page 4 SOLUTION: greater than 1, so the graph is a vertical stretch of . Another way to identify the stretch is to notice that the y-values in the table are 7 times the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}. identify the compression is to notice that the y-values in the table are times the corresponding y-values 10-1for Square Rootfunction. Functions the parent The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}. 16. 18. SOLUTION: x 0 0.5 y 0 ≈ – 0.2 1 –0.3 2 ≈ – 0.5 3 ≈ – 0.6 SOLUTION: x 0 0.5 y 0 ≈ – 0.2 4 ≈ – 0.7 1 – 0.25 2 ≈ – 0.35 3 ≈ – 0.4 4 –0.5 The parent function is multiplied by a value less than 1, so the graph is a vertical compression of and a reflection across the x-axis. Another way to identify the compression is to notice that the The parent function is multiplied by a value less than 1, so the graph is a vertical compression of and a reflection across the x-axis. Another way to identify the compression is to notice that the y-values in the table are times the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}. y-values in the table are times the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}. 17. 19. SOLUTION: x 0 0.5 y 0 ≈ 4.9 1 7 2 ≈ 9.9 3 ≈ 12.1 SOLUTION: x 0 0.5 y 0 ≈ – 0.7 4 14 The parent function is multiplied by a value greater than 1, so the graph is a vertical stretch of . Another way to identify the stretch is to notice that the y-values in the table are 7 times the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}. 1 –1 The parent function 2 ≈ – 1.4 3 ≈ – 1.7 4 –2 is multiplied by the negative of 1, so the graph is a reflection of across the x-axis. Another way to identify the graph is to notice that the y-values in the table are –1 times the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}. 18. 20. eSolutions Manual - Powered by Cognero SOLUTION: x 0 0.5 y 0 ≈ – 1 – Page 5 2 ≈ – 3 ≈ – 4 –0.5 SOLUTION: x 0 0.5 1 2 3 4 negative of 1, so the graph is a reflection of across the x-axis. Another way to identify the graph is to notice that the y-values in the table are –1 times 10-1the Square Root Functions corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}. and a reflection across the x-axis. Another way to identify the stretch is to notice that the yvalues in the table are –7 times the corresponding yvalues for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}. 22. 20. SOLUTION: x 0 0.5 y 0 ≈ – 0.1 1 –0.2 2 ≈ – 0.3 3 ≈ – 0.35 SOLUTION: x 0 0.5 y 2 ≈ 2.7 4 –0.4 1 3 2 ≈ 3.4 3 ≈ 3.7 4 4 The value 2 is being added to the parent function , so the graph is translated up 2 units from The parent function is multiplied by a value less than 1, so the graph is a vertical compression of and a reflection across the x-axis. Another way to identify the compression is to notice that the y-values in the table are times the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}. the parent graph . Another way to identify the translation is to note that the y-values in the table are 2 greater than the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 2}. 23. SOLUTION: x 0 0.5 y 4 ≈ 4.7 21. SOLUTION: x 0 0.5 y 0 ≈ – 4.9 1 –7 2 ≈ – 9.9 3 ≈ – 12.1 1 5 2 ≈ 5.4 3 ≈ 5.7 4 6 4 –14 The value 4 is being added to the parent function , so the graph is translated up 4 units from The parent function is multiplied by a value less than 1, so the graph is a vertical stretch of and a reflection across the x-axis. Another way to identify the stretch is to notice that the yvalues in the table are –7 times the corresponding yvalues for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}. eSolutions Manual - Powered by Cognero 22. SOLUTION: the parent graph . Another way to identify the translation is to note that the y-values in the table are 4 greater than the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 4}. 24. SOLUTION: x 0 0.5 y –1 ≈ – 0.3 1 0 2 ≈ 0.4 3 ≈ 0.7 4 Page 1 6 the parent graph . Another way to identify the translation is to note that the y-values in the table are 4 greater than the corresponding y-values for the 10-1parent Square Root Functions function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 4}. 24. units from the parent graph . Another way to identify the translation is to note that the y-values in the table are 3 less than the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ –3}. 26. SOLUTION: x 0 0.5 y –1 ≈ – 0.3 1 0 2 ≈ 0.4 3 ≈ 0.7 SOLUTION: x 0 0.5 y 1.5 ≈ 2.2 4 1 1 2.5 2 ≈ 2.9 3 ≈ 3.2 4 3.5 The value 1.5 is being added to the parent function , so the graph is translated up 1.5 units from The value 1 is being subtracted from the parent function , so the graph is translated down 1 the parent graph . Another way to identify the translation is to note that the y-values in the table are 1.5 greater than the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 1.5}. unit from the parent graph . Another way to identify the translation is to note that the y-values in the table are 1 less than the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ –1}. 27. 25. SOLUTION: x 0 0.5 y –3 ≈ – 2.3 1 –2 2 ≈ – 1.6 3 ≈ – 1.3 SOLUTION: x 0 1 y –2.5 –1.5 4 –1 2 –1.1 4 –0.5 6 ≈ – 0.1 6.5 ≈ 0 The value 2.5 is being subtracted from the parent function , so the graph is translated down 2.5 The value 3 is being subtracted from the parent function , so the graph is translated down 3 units from the parent graph . Another way to identify the translation is to note that the y-values in the table are 2.5 less than the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ –2.5}. units from the parent graph . Another way to identify the translation is to note that the y-values in the table are 3 less than the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ –3}. 28. 26. eSolutions Manual - Powered by Cognero SOLUTION: x y 0 1.5 0.5 ≈ 2.2 1 2.5 2 ≈ 2.9 3 ≈ 3.2 4 3.5 SOLUTION: x –4 –3 y 0 1 –2 ≈ 1.4 0 2 2 ≈ 2.4 Page 4 7 ≈ 2.8 translated 4 units right from the parent graph . Another way to identify the translation is to note that the x-values in the table are 4 more than the corresponding x-values for the parent function. The domain is {x|x ≥ 4}, and the range is {y|y ≥ 0}. units from the parent graph . Another way to identify the translation is to note that the y-values in the table are 2.5 less than the corresponding y-values 10-1for Square Rootfunction. Functions the parent The domain is {x|x ≥ 0} and the range is {y|y ≥ –2.5}. 28. 30. SOLUTION: x –4 –3 y 0 1 –2 ≈ 1.4 0 2 2 ≈ 2.4 SOLUTION: x –1 0 y 0 1 4 ≈ 2.8 1 ≈ 1.4 2 ≈ 1.7 3 2 5 ≈ 2.4 The value 4 is being added to the square root of the parent function , so the graph is translated 4 The value 1 is being added to the square root of the parent function , so the graph is translated 1 units left from the parent graph . Another way to identify the translation is to note that the xvalues in the table are 4 less than the corresponding x-values for the parent function. The domain is {x|x ≥ –4}, and the range is {y|y ≥ 0}. unit left from the parent graph . Another way to identify the translation is to note that the x-values in the table are 1 less than the corresponding xvalues for the parent function. The domain is {x|x ≥ –1}, and the range is {y|y ≥ 0}. 29. 31. SOLUTION: x 4 5 y 0 1 6 ≈ 1.4 6.5 ≈ 1.6 7 ≈ 1.7 SOLUTION: x 0.5 1 y 0 ≈ 0.7 8 2 1.5 1 2 ≈ 1.2 3 ≈ 1.6 4.5 2 The value 0.5 is being subtracted from the square root of the parent function , so the graph is translated 0.5 unit right from the parent graph . Another way to identify the translation is to note that the x-values in the table are 0.5 more than the corresponding x-values for the parent function. The domain is {x|x ≥ 0.5}, and the range is {y|y ≥ 0}. The value 4 is being subtracted from the square root of the parent function , so the graph is translated 4 units right from the parent graph . Another way to identify the translation is to note that the x-values in the table are 4 more than the corresponding x-values for the parent function. The domain is {x|x ≥ 4}, and the range is {y|y ≥ 0}. 32. 30. eSolutions Manual - Powered by Cognero SOLUTION: x y –1 0 0 1 1 ≈ 2 ≈ 1.7 3 2 5 ≈ SOLUTION: x –5 –4 y 0 1 –1 2 0 ≈ 2.2 2 ≈ 2.6 Page 8 4 3 translated 0.5 unit right from the parent graph . Another way to identify the translation is to note that the x-values in the table are 0.5 more than 10-1the Square Root Functions corresponding x-values for the parent function. The domain is {x|x ≥ 0.5}, and the range is {y|y ≥ 0}. 34. GEOMETRY The perimeter of a square is given by the function , where A is the area of the square. a. Graph the function. b. Determine the perimeter of a square with an area 32. SOLUTION: x –5 –4 y 0 1 note that the x-values in the table are 1.5 more than the corresponding x-values for the parent function. The domain is {x|x ≥ 1.5}, and the range is {y|y ≥ 0}. –1 2 0 ≈ 2.2 2 ≈ 2.6 4 3 2 of 225 m . c. When will the perimeter and the area be the same value? SOLUTION: a. Make a table of values. Use perfect squares for A to make easy to calculate. A 4 9 16 25 P 8 12 16 20 The value 5 is being added to the square root of the parent function , so the graph is translated 5 units left from the parent graph . Another way to identify the translation is to note that the xvalues in the table are 5 less than the corresponding x-values for the parent function. The domain is {x|x ≥ –5}, and the range is {y|y ≥ 0}. 33. b. SOLUTION: x 1.5 2.5 y 0 1 3 ≈ 1.2 4 ≈ 1.6 5 ≈ 1.9 5.5 2 The perimeter of the square would be 60m. c. Set A = P, and substitute P for A. Then solve for P. The value 1.5 is being subtracted from the square root of the parent function , so the graph is translated 1.5 units right from the parent graph . Another way to identify the translation is to note that the x-values in the table are 1.5 more than the corresponding x-values for the parent function. The domain is {x|x ≥ 1.5}, and the range is {y|y ≥ 0}. 34. GEOMETRY The perimeter of a square is given by the function , where A is the area of the square. eSolutions Manual - Powered by Cognero a. Graph the function. b. Determine the perimeter of a square with an area 2 The perimeter and area would be the same value 2 when the area is 16 m , the perimeter is 16 m, or the length of the sides of the square are 4 m. Page 9 Graph each function, and compare to the parent graph. State the domain and range. This graph is the result of a vertical stretch of the 10-1 Square Root Functions The perimeter and area would be the same value 2 when the area is 16 m , the perimeter is 16 m, or the length of the sides of the square are 4 m. graph of y = followed by a reflection across the x-axis, and then a translation 3 units down. The domain is {x|x ≥ 0}, and the range is {y|y ≤ –3}. 37. Graph each function, and compare to the parent graph. State the domain and range. SOLUTION: x –2 –1 y 0 0.5 35. SOLUTION: x 0 0.5 y 2 ≈ 0.6 1 0 2 ≈ – 0.8 3 ≈ – 1.5 0 ≈ 0.7 1 ≈ 0.9 2 1 4 ≈ 1.2 4 –2 This graph is the result of a vertical compression of the graph of y = followed by a translation 2 units left. The domain is {x|x ≥ –2}, and the range is {y|y ≥ 0}. This graph is the result of a vertical stretch of the graph of y = followed by a reflection across the x-axis, and then a translation 2 units up. The domain is {x|x ≥ 0}, and the range is {y|y ≤ 2}. 38. SOLUTION: x 1 2 y 0 –1 36. SOLUTION: x 0 0.5 y –3 ≈ – 5.1 1 –6 2 ≈ – 7.2 3 ≈ – 8.2 graph of y = followed by a reflection across the x-axis, and then a translation 3 units down. The domain is {x|x ≥ 0}, and the range is {y|y ≤ –3}. 4 ≈ – 1.7 5 –2 This graph is the result of a reflection across the xaxis of the graph of followed by a translation 1 unit right. The domain is {x|x ≥ 1}, and the range is {y|y ≤ 0}. 39. SOLUTION: x 1 2 y 2 2.25 37. 2 1 3.5 ≈ – 1.6 4 –9 This graph is the result of a vertical stretch of the SOLUTION: x –2 –1 0 1 eSolutions Manual - Powered by Cognero y 0 0.5 ≈ ≈ 0.9 0.7 3 ≈ – 1.4 4 ≈ 1.2 3 ≈ 2.35 4 ≈ 2.4 5 2.5 10 2.75 Page 10 This graph is the result of a reflection across the xaxis of the graph of followed by a translation 10-11Square Root unit right. TheFunctions domain is {x|x ≥ 1}, and the range is {y|y ≤ 0}. This graph is the result of a vertical compression of the graph of followed by a translation 1 unit up and 2 units right. The domain is {x|x ≥ 2}, and the range is {y|y ≥ 1}. 41. ENERGY An object has kinetic energy when it is in motion. The velocity in meters per second of an object of mass m kilograms with an energy of E 39. SOLUTION: x 1 2 y 2 2.25 3 ≈ 2.35 4 ≈ 2.4 5 2.5 10 2.75 joules is given by the function . Use a graphing calculator to graph the function that represents the velocity of a basketball with a mass of 0.6 kilogram. SOLUTION: Use a graphing calculator to graph the function where m = 0.6. This graph is the result of a vertical compression of the graph of followed by a translation 2 units up and 1 unit right. The domain is {x|x ≥ 1}, and the range is {y|y ≥ 2}. 40. SOLUTION: x 2 3 y 1 1.5 4 ≈ 1.6 5 ≈ 1.9 6 2 7 ≈ 2.1 42. GEOMETRY The radius of a circle is given by , where A is the area of the circle. a. Graph the function. b. Use a graphing calculator to determine the radius 2 of a circle that has an area of 27 in . This graph is the result of a vertical compression of the graph of followed by a translation 1 unit up and 2 units right. The domain is {x|x ≥ 2}, and the range is {y|y ≥ 1}. 41. ENERGY An object has kinetic energy when it is in motion. The velocity in meters per second of an object of mass m kilograms with an energy of E joules is given by the function SOLUTION: r 0 1 7 10 ≈1.5 ≈1.8 . Use a graphing calculator to graph the function that with a mass of 0.6 kilogram. represents velocity of a basketball eSolutions Manual -the Powered by Cognero SOLUTION: a. A 0 3.14 Page 11 43. SPEED OF SOUND The speed of sound in air is determined by the temperature of the air. The speed c in meters per second is given by c = 10-1 Square Root Functions 331.5 42. GEOMETRY The radius of a circle is given by , where A is the area of the circle. a. Graph the function. b. Use a graphing calculator to determine the radius , where t is the temperature of the air in degrees Celsius. a. Use a graphing calculator to graph the function. b. How fast does sound travel when the temperature is 55°C? c. How is the speed of sound affected when the temperature increases to 65°C? Explain. SOLUTION: a. 2 of a circle that has an area of 27 in . SOLUTION: a. A 0 3.14 r 0 1 7 10 ≈1.5 ≈1.8 b. b. Use a graphing calculator to graph . [0, 1000] scl: 20 by [0, 1000] scl: 10 Use the value option from the CALC menu to find the value or r at A = 27. [-2, 28] scl: 3 by [-2, 8] scl: 1 When the temperature is 55°C, sound travels at about 363.3 m/s. c. At A = 27, r = 2.9. The radius of a circle that has an 2 area of 27 in is about 2.9 in. 43. SPEED OF SOUND The speed of sound in air is determined by the temperature of the air. The speed c in meters per second is given by c = 331.5 , where t is the temperature of the air inManual degrees Celsius. eSolutions - Powered by Cognero a. Use a graphing calculator to graph the function. b. How fast does sound travel when the [0, 1000] scl: 20 by [0, 1000] scl: 10 When t is 65°C, c is about 368.8 m/s, so this 10degree increase results in an increase in speed of about 5.5 m/s. Page 12 44. MULTIPLE REPRESENTATIONS In this problem, you will explore the relationship between 1000]Root scl: 20 by [0, 1000] scl: 10 10-1[0, Square Functions When t is 65°C, c is about 368.8 m/s, so this 10degree increase results in an increase in speed of about 5.5 m/s. 44. MULTIPLE REPRESENTATIONS In this problem, you will explore the relationship between the graphs of square root functions and parabolas. 2 a. GRAPHICAL Graph y = x on a coordinate system. b. ALGEBRAIC Write a piecewise-defined 2 function to describe the graph of y = x in each quadrant. c. GRAPHICAL On the same coordinate system, graph . and d. GRAPHICAL On the same coordinate system, graph the line y = x. Plot the points (2, 4), (4, 2), and (1, 1). e. ANALYTICAL Compare the graph of the parabola to the graphs of the square root functions. SOLUTION: a. d. e . The combined graphs of the square root functions are the same size and shape as the parabola. They are a reflection of the parabola across the line y = x. Point (4, 2) is reflected to (2, 4). The point (1, 1) is on both functions. CHALLENGE Determine whether each statement is true or false . Provide an example or counterexample to support your answer. 45. Numbers in the domain of a radical function will always be nonnegative. SOLUTION: Numbers in the domain of a radical function will not always be nonnegative.For example the domain of is {x|x ≥ −3}. b. In the first quadrant, the function is the second quadrant, the function is Combine the two to write a piecewise-defined function . In . 46. Numbers in the range of a radical function will always be nonnegative. SOLUTION: Numbers in the range of a radical function will not always be nonnegative.For example, −6 and −5 are in the range of . 47. WRITING IN MATH Why are there limitations on the domain and range of square root functions? c. SOLUTION: is not a real number. d. eSolutions Manual - Powered by Cognero The term underneath the square root cannot be negative, which means the domain must be restricted. The square root function will also never attain a value less than 0, which means that the range is restricted. Page 13 48. CCSS TOOLS Write a radical function with a domain of all real numbers greater than or equal to 2 SOLUTION: Numbers in the range of a radical function will not always be nonnegative.For example, −6 and −5 are in 10-1 Square Root Functions the range of . 47. WRITING IN MATH Why are there limitations on the domain and range of square root functions? be all real numbers greater than or equal to 2 because there cannot be a negative number under the radicand. The range is all real numbers less than or equal to 5 because the function is translated 5 units up, but then reflected across the x-axis. 49. WHICH DOES NOT BELONG? Identify the equation that does not belong. Explain. SOLUTION: is not a real number. SOLUTION: does not belong because it is a translation The term underneath the square root cannot be negative, which means the domain must be restricted. The square root function will also never attain a value less than 0, which means that the range is restricted. of . The other equations represent vertical stretches or compressions since a each is multiplied by a constant. 48. CCSS TOOLS Write a radical function with a domain of all real numbers greater than or equal to 2 and a range of all real numbers less than or equal to 5. SOLUTION: Sample answer: . The domain must be all real numbers greater than or equal to 2 because there cannot be a negative number under the radicand. The range is all real numbers less than or equal to 5 because the function is translated 5 units up, but then reflected across the x-axis. 49. WHICH DOES NOT BELONG? Identify the equation that does not belong. Explain. [-2, 18] scl: 2 by [-2, 8] scl: 1 50. OPEN ENDED Write a function that is a reflection, translation, and a dilation of the parent graph . SOLUTION: Multiplying by –1 reflects the function. Multiplying by 3 dilates the function. Subtracting 1 from x translates the function. SOLUTION: does not belong because it is a translation of . The other equations represent vertical stretches or compressions since a each is multiplied by a constant. [-2, 18] scl; 2 by [-1, 10] scl: 2 51. REASONING If the range of the function y = a is {y|y ≤ 0}, what can you conclude about the value of a? Explain your reasoning. SOLUTION: The value of a is negative. For the function to have negative y-values, the value of a must be negative. For example has the range of {y|y ≤ 0} eSolutions Manual - Powered by Cognero [-2, 18] scl: 2 by [-2, 8] scl: 1 Page 14 52. WRITING IN MATH Compare and contrast the graphs of f (x) = and . 10-1 Square Root Functions [-2, 18] scl; 2 by [-1, 10] scl: 2 51. REASONING If the range of the function y = a is {y|y ≤ 0}, what can you conclude about the value of a? Explain your reasoning. 53. SOLUTION: The value of a is negative. For the function to have negative y-values, the value of a must be negative. For example has the range of {y|y ≤ 0} 52. WRITING IN MATH Compare and contrast the graphs of f (x) = and . SOLUTION: Both functions are translations of the square root function. f (x) = is a translation 2 units up. g (x) = is a translation 2 units to the left. When the constant is added or subtracted to x under the radical, the function is translated left or right. If the constant is added or subtracted outside the radical, the square root function is translated up or down. 53. Which function best represents the graph? 2 A y = x B y = 2x C y = D y = x SOLUTION: Choice B is an exponential function, which is not represented by the graph. Choice C is a square root function, which is not represented by the graph. Choice D is a linear function, which is not represented by the graph. Choice A is a quadratic function, which graphs as a parabola. The correct choice is A. 54. The statement “x < 10 and 3x − 2 ≥ 7” is true when x is equal to F 0 G 2 H 8 J 12 SOLUTION: Which function best represents the graph? 2 A y = x B y = 2x C y = D y = x SOLUTION: Choice B is an exponential function, which is not represented by the graph. Choice C is a square root function, which is not represented by the graph. Choice D is a linear function, which is not eSolutions Manual - Powered by Cognero represented by the graph. Choice A is a quadratic function, which graphs as a parabola. So, 3 ≤ x < 10. Neither 0 nor 2 are greater than 3, and 12 is not less than 10. Eight is both greater than 3 and less than 10. The correct choice is H. 55. Which of the following is the equation of a line parallel to and passing through (−2, −1)? A y = B y = 2x + 3 C Page 15 D So, 3 ≤ x < 10. Neither 0 nor 2 are greater than 3, and 12 is not less than 10. Eight is both greater than 3 less than 10-1and Square Root10. Functions The correct choice is H. 55. Which of the following is the equation of a line parallel to and passing through (−2, Therefore, 13 of mulch are needed to cover the flower beds. Graph each function. 57. f (x) = | 3x + 2 | SOLUTION: Since f (x) cannot be negative, the minimum point of the graph is where f (x) = 0. −1)? A y = B y = 2x + 3 C D Make a table of values x 0 –2 −1 f (x) 4 1 2 SOLUTION: Parallel lines have the same slope. Substitute the point and slope into the point-slope form of a linear equation. The correct choice is D. 1 5 2 8 58. f (x) = 56. SHORT RESPONSE A landscaper needs to mulch 6 rectangular flower beds that are 8 feet by 4 feet and 4 circular flower beds each with a radius of 3 feet. One bag of mulch covers 25 square feet. How many bags of mulch are needed to cover the flower beds? SOLUTION: This is a piecewise-defined function. Make a table of values. Be sure to include the domain values for which the function changes. x 2 3 –3 −2 –1 f (x) 0 1 2 0 1 Notice that both functions are linear. SOLUTION: 59. Therefore, 13 of mulch are needed to cover the flower beds. Graph each function. 57. f (x) = | 3x + 2 | SOLUTION: Since f (x) cannot bebynegative, eSolutions Manual - Powered Cognero the minimum point of the graph is where f (x) = 0. SOLUTION: Make a table of values. x f (x) –5 –4 –4 –3 –3 –2 –2 –1 0 –1 0 1 Page 16 10-1 Square Root Functions 59. 60. f (x) = SOLUTION: Make a table of values. x f (x) –5 –4 –4 –3 –3 –2 –2 –1 0 –1 0 1 1 2 2 3 3 4 4 5 SOLUTION: Since f (x) cannot be negative, the minimum point of the graph is where f (x) = 0. Make a table of values x 0 2 –2 f (x) 1 4 0 6 60. f (x) = SOLUTION: Since f (x) cannot be negative, the minimum point of the graph is where f (x) = 0. Graph each set of ordered pairs. Determine whether the ordered pairs represent a linear function, a quadratic function, or an exponential function. 61. {(−2, 5), (−1, 3), (0, 1), (1, −1), (2, −3)} SOLUTION: Make a table of values x 0 2 –2 f (x) 1 4 0 6 The points can be connected by a straight line. Thus, the ordered pairs represent a linear function. 62. {(0, 0), (1, 3), (2, 4), (3, 3), (4, 0)} eSolutions Manual - Powered by Cognero Graph each set of ordered pairs. Determine whether the ordered pairs represent a linear SOLUTION: Page 17 The points can be connected by a straight line. Thus, the ordered pairs represent a linear function. 10-1 Square Root Functions 62. {(0, 0), (1, 3), (2, 4), (3, 3), (4, 0)} A curve can be drawn with the given points. The curve has no line of symmetry and is rapidly decreasing. Thus, the ordered pairs represent an exponential function. 64. {(−4, 4), (−2, 1), (0, 0), (2, 1), (4, 4)} SOLUTION: SOLUTION: A parabola can be drawn with the given points. The curve has a line of symmetry and points are reflected over this line. Thus, the ordered pairs represent a quadratic function. A parabola can be drawn with the given points. The curve has a line of symmetry and points are reflected over this line. Thus, the ordered pairs represent a quadratic function. 63. SOLUTION: A curve can be drawn with the given points. The curve has no line of symmetry and is rapidly decreasing. Thus, the ordered pairs represent an exponential function. 64. {(−4, 4), (−2, 1), (0, 0), (2, 1), (4, 4)} SOLUTION: 65. HEALTH Aida exercises every day by walking and jogging at least 3 miles. Aida walks at a rate of 4 miles per hour and jogs at a rate of 8 miles per hour. Suppose she has at most one half-hour to exercise today. a. Draw a graph showing the possible amounts of time she can spend walking and jogging today. b. List three possible solutions. SOLUTION: a. Let x be the number of minutes Aida walks and y be the number of minutes Aida jogs. If Aida can spend no more than one-half hour exercising, then the sum of the number of minutes she walks and jogs must be less than or equal to 30. This is expressed by the following inequality. Since the time is represented in minutes, Aida's rate walking and jogging must be converted from miles per hour to miles per minute by dividing each by 60. Her rate walking is miles per minute and her rate jogging is miles per minute. The following inequality that uses the distance formula (d = rt), represents the fact that the distance Aida walks plus the distance she jogs is greater than or equal to a total distance of 3 miles. A parabola can be drawn with the given points. The curve has a line of symmetry and points are reflected eSolutions - Powered Cognero Thus,bythe ordered pairs represent a overManual this line. function. quadratic Page 18 her rate jogging is miles per minute. The following inequality that uses the distance formula (d rt), represents the fact that the distance Aida 10-1=Square Root Functions walks plus the distance she jogs is greater than or equal to a total distance of 3 miles. SOLUTION: The graph shows a positive correlation because as the number of grams of fat increases, the amount of Calories also increases. Factor each monomial completely. 67. 28n 3 SOLUTION: Graph the two inequalities on the same coordinate plane and find their intersection. 2 68. −33a b SOLUTION: 69. 150rt SOLUTION: 2 2 70. −378nq r b. In order to walk and jog at least 3 miles and exercise no more than 30 minutes, Aida could: walk for 15 minutes and jog for 15 minutes; walk for 10 minutes and jog for 18 minutes; walk for 5 minutes and jog for 23 minutes. 66. NUTRITION Determine whether the graph shows a positive , negative, or no correlation. If there is a positive or negative correlation, describe its meaning in the situation. SOLUTION: 3 2 71. 225a b c SOLUTION: 2 4 72. −160x y SOLUTION: SOLUTION: The graph shows a positive correlation because as the number of grams of fat increases, the amount of Calories also increases. Factor each monomial completely. 3 eSolutions 67. 28n Manual - Powered by Cognero SOLUTION: Page 19 10-2 Simplifying Radical Expressions Simplify each expression. 6. 1. SOLUTION: SOLUTION: 7. SOLUTION: 2. SOLUTION: 8. SOLUTION: 3. SOLUTION: 9. SOLUTION: 4. SOLUTION: 10. MULTIPLE CHOICE Which expression is equivalent to 5. ? A SOLUTION: B C D 6. SOLUTION: eSolutions Manual - Powered by Cognero 7. SOLUTION: Page 1 SOLUTION: 10-2 Simplifying Radical Expressions 10. MULTIPLE CHOICE Which expression is equivalent to 12. ? SOLUTION: A B C D 13. SOLUTION: SOLUTION: 14. The correct choice is D. SOLUTION: Simplify each expression. 11. SOLUTION: 15. SOLUTION: 12. SOLUTION: eSolutions Manual - Powered by Cognero 13. 16. Page 2 SOLUTION: 10-2 Simplifying Radical Expressions 20. 16. SOLUTION: SOLUTION: 21. SOLUTION: Simplify each expression. 17. SOLUTION: 22. SOLUTION: 18. SOLUTION: 23. SOLUTION: 19. SOLUTION: 24. SOLUTION: 20. SOLUTION: 25. SOLUTION: eSolutions Manual - Powered by Cognero 21. Page 3 26. SOLUTION: 10-2 Simplifying Radical Expressions 25. 30. SOLUTION: SOLUTION: 31. 26. SOLUTION: SOLUTION: 32. 27. SOLUTION: SOLUTION: 33. SOLUTION: 28. SOLUTION: 34. SOLUTION: 29. SOLUTION: 35. ROLLER COASTER Starting from a stationary position, the velocity v of a roller coaster in feet per second at the bottom of a hill can be approximated by , where h is the height of the hill in feet. 30. SOLUTION: eSolutions Manual - Powered by Cognero 31. a. Simplify the equation. b. Determine the velocity of a roller coaster at the bottom of a 134-foot hill. SOLUTION: a. Page 4 SOLUTION: pump that is advertised to project water with a velocity of 77 feet per second meet the fire department’s need? Explain. 10-2 Simplifying Radical Expressions SOLUTION: a. 35. ROLLER COASTER Starting from a stationary position, the velocity v of a roller coaster in feet per second at the bottom of a hill can be approximated by , where h is the height of the hill in feet. a. Simplify the equation. b. Determine the velocity of a roller coaster at the bottom of a 134-foot hill. b. To determine the height of the water, substitute 70 for v in the function SOLUTION: a. . A pump with a velocity of 70 feet per second will pump water only to a maximum height of 76.6 feet. Therefore, this pump will not meet the fire department’s need. c. To determine the height of the water, substitute 77 b. To determine the velocity of the roller coaster at the bottom of the hill, substitute 134 for h in the equation . for v in the function . The roller coaster will have a velocity of about 92.6 ft/sec at the bottom of a 134-foot hill. 36. CCSS PRECISION When fighting a fire, the velocity v of water being pumped into the air is A pump with a velocity of 77 feet per second will pump water to a maximum height of 92.6 feet. Therefore, this pump will meet the fire department’s need. modeled by the function , where h represents the maximum height of the water and g 2 represents the acceleration due to gravity (32 ft/s ). a. Solve the function for h. b. The Hollowville Fire Department needs a pump that will propel water 80 feet into the air. Will a pump advertised to project water with a velocity of 70 feet per second meet their needs? Explain. c. The Jackson Fire Department must purchase a pump that will propel water 90 feet into the air. Will a pump that is advertised to project water with a velocity of 77 feet per second meet the fire department’s need? Explain. Simplify each expression. 37. SOLUTION: SOLUTION: a. eSolutions Manual - Powered by Cognero b. To determine the height of the water, substitute 70 Page 5 A pump with a velocity of 77 feet per second will pump water to a maximum height of 92.6 feet. 10-2Therefore, Simplifying Expressions thisRadical pump will meet the fire department’s need. Simplify each expression. 39. 37. SOLUTION: SOLUTION: 40. SOLUTION: 38. SOLUTION: 41. SOLUTION: 39. SOLUTION: 42. eSolutions Manual - Powered by Cognero SOLUTION: Page 6 10-2 Simplifying Radical Expressions 42. 45. SOLUTION: SOLUTION: 46. 43. SOLUTION: SOLUTION: 47. SOLUTION: 44. SOLUTION: 48. SOLUTION: 45. SOLUTION: 49. ELECTRICITY The amount of current in amperes I that an appliance uses can be calculated using the eSolutions Manual - Powered by Cognero Page 7 formula , where P is the power in watts and R is the resistance in ohms. 10-2 Simplifying Radical Expressions An appliance uses about 3.9 amps of current if the power used is 75 watts and the resistance is 5 ohms. 49. ELECTRICITY The amount of current in amperes I that an appliance uses can be calculated using the 50. KINETIC ENERGY The speed v of a ball can be determined by the equation formula , where P is the power in watts and R is the resistance in ohms. a. Simplify the formula. b. How much current does an appliance use if the power used is 75 watts and the resistance is 5 ohms? SOLUTION: a. , where k is the kinetic energy and m is the mass of the ball. a. Simplify the formula if the mass of the ball is 3 kilograms. b. If the ball is traveling 7 meters per second, what is the kinetic energy of the ball in Joules? SOLUTION: a. If the mass of the ball is 3 kilograms, substitute m = 3 into the equation. b. Substitute P = 75 and R = 5 in the equation . b. Substitute V = 7 in the equation . An appliance uses about 3.9 amps of current if the power used is 75 watts and the resistance is 5 ohms. 50. KINETIC ENERGY The speed v of a ball can be determined by the equation , where k is the kinetic energy and m is the mass of the ball. a. Simplify the formula if the mass of the ball is 3 kilograms. b. If the ball is traveling 7 meters per second, what is the kinetic energy of the ball in Joules? SOLUTION: a. If the mass of the ball is 3 kilograms, substitute m = 3 into the equation. The kinetic energy of the ball is 73.5 Joules. 51. SUBMARINES The greatest distance d in miles th the lookout can see on a clear day is modeled by the formula . Determine how high the submarine would have to raise its periscope to see a ship, if the submarine is the given distances away fro a ship. eSolutions Manual - Powered by Cognero Page 8 = d (3) 2 10-2 Simplifying Radical Expressions The kinetic energy of the ball is 73.5 Joules. 2 =6 = 24 (9) 2 = 54 2 (12) = 96 2 (15) = 150 51. SUBMARINES The greatest distance d in miles th the lookout can see on a clear day is modeled by the formula (6) 52. CCSS STRUCTURE Explain how to solve . . Determine how high the submarine would have to raise its periscope to see a ship, if the submarine is the given distances away fro a ship. SOLUTION: To solve an equation of equal ratios, first find the equal cross products and then solve for the variable. Use the conjugate of denominator. to rationalize the SOLUTION: First solve the equation for h. So, the solution is Distance Height = d 3 h= 2 6 h= (3) 2 =6 9 h= (6) 2 = 24 (9) 2 = 54 12 15 h= h= 2 (12) = 96 2 (15) = 150 52. CCSS STRUCTURE Explain how to solve . SOLUTION: To solve an equation of equal ratios, first find the equal cross products and then solve for the variable. eSolutions Manual - Powered by Cognero . 53. CHALLENGE Simplify each expression. a. b. c. SOLUTION: a. b. c. 54. REASONING Marge takes a number, subtracts 4, multiplies by 4, takes the square root, and takes the reciprocal to get . What number did she start with? Write a formula to describe the process. SOLUTION: Page 9 b. 10-2 Simplifying Radical Expressions c. are 54. REASONING Marge takes a number, subtracts 4, multiplies by 4, takes the square root, and takes the and . 56. CHALLENGE Use the Quotient Property of Square Roots to derive the Quadratic Formula by 2 reciprocal to get . What number did she start with? Write a formula to describe the process. solving the quadratic equation ax + bx + c = 0. (Hint: Begin by completing the square.) SOLUTION: SOLUTION: Let x = a number. 57. WRITING IN MATH Summarize how to write a radical expression in simplest form. 55. OPEN ENDED Write two binomials of the form and . Then find their product. SOLUTION: and Two binomials of the form are and SOLUTION: No radicals can appear in the denominator of a fraction. So, rationalize the denominator to get rid of the radicand in the denominator. Then check if any of the radicands have perfect square factors other than 1. If so, simplify. For example, simplify the following. . 56. CHALLENGE Use the Quotient Property of Square Roots to derive the Quadratic Formula by 2 solving the quadratic equation ax + bx + c = 0. (Hint: Begin by completing the square.) SOLUTION: eSolutions Manual - Powered by Cognero 58. Jerry’s electric bill is $23 less than his natural gas bill. The two bills are a total of $109. Which of the following equations can be used to find the amount of his natural gas bill? A g + g = 109 B 23 + 2g = 109 C g − 23 = 109 D 2g − 23 = 109 SOLUTION: Let g = Jerry’s natural gas bill. Jerry’s electric bill is $23 less than his natural gas bill, so the electric bill is g – 23. The two bills are a total of $109. Page 10 10-2 Simplifying Radical Expressions The roots are –4 and 6. So, the correct choice is H. 58. Jerry’s electric bill is $23 less than his natural gas bill. The two bills are a total of $109. Which of the following equations can be used to find the amount of his natural gas bill? A g + g = 109 B 23 + 2g = 109 C g − 23 = 109 D 2g − 23 = 109 60. The expression the following? A is equivalent to which of B C D SOLUTION: Let g = Jerry’s natural gas bill. Jerry’s electric bill is $23 less than his natural gas bill, so the electric bill is g – 23. The two bills are a total of $109. SOLUTION: So, the correct choice is D. So, the correct choice is C. 2 59. Solve a − 2a + 1 = 25. F −4, −6 G 4, −6 H −4, 6 J 4, 6 61. GRIDDED RESPONSE Miki earns $10 an hour and 10% commission on sales. If Miki worked 38 hours and had a total sales of $1275 last week, how much did she make? SOLUTION: Miki’s earnings are $10 per hour and 10% commission on sales. SOLUTION: Solve for a. Miki made $507.50 last week. Graph each function. Compare to the parent graph. State the domain and range. 62. The roots are –4 and 6. So, the correct choice is H. 60. The expression the following? A SOLUTION: x 0 1 y 1 –1 2 ≈ 1.8 3 ≈ 2.5 4 3 is equivalent to which of B C D SOLUTION: eSolutions Manual - Powered by Cognero is multiplied by a value The parent function greater than 1 and is subtracted by the value 1, so the Page 11 graph is a vertical stretch of followed by a translation 1 unit down. The domain is {x|x ≥ 0}, and the range is {y|y ≥ -1}. identify the compression is to notice that the y-values in the table are times the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}. 10-2 Simplifying Radical Expressions Miki made $507.50 last week. Graph each function. Compare to the parent graph. State the domain and range. 64. SOLUTION: x –2 –1 y 0 2 62. SOLUTION: x 0 1 y 1 –1 2 ≈ 1.8 3 ≈ 2.5 0 ≈ 2.8 1 ≈ 3.5 2 4 4 3 This graph is the result of a vertical stretch of the is multiplied by a value The parent function greater than 1 and is subtracted by the value 1, so the graph is a vertical stretch of followed by a translation 1 unit down. The domain is {x|x ≥ 0}, and the range is {y|y ≥ -1}. graph of y = followed by a translation 2 units left. The domain is {x|x ≥ –2}, and the range is {y|y ≥ 0}. 65. SOLUTION: x 0 –1 y 0 –1 63. SOLUTION: x 0 1 y 0 2 ≈ 0.7 3 ≈0 .9 1 ≈ – 1.4 2 ≈ – 1.7 4 1 This graph is the result of a reflection across the xaxis of the graph of followed by a translation 1 unit left. The domain is {x|x ≥ –1}, and the range is {y|y ≤ 0}. The parent function is multiplied by a value less than 1 and greater than 0, so the the graph is a vertical compression of . Another way to identify the compression is to notice that the y-values in the table are times the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}. 64. eSolutions Manual - Powered by Cognero SOLUTION: x –2 –1 0 66. SOLUTION: x 3 4 y 0 –3 5 ≈ – 4.2 6 ≈ – 5.2 Page 12 1 2 This graph is the result of a reflection across the xaxis of the graph of followed by a translation unit left. TheRadical domain is {x|x ≥ –1}, and the range is 10-21Simplifying Expressions {y|y ≤ 0}. Determine the domain and range for each function. 68. f (x) = |2x − 5| 66. SOLUTION: x 3 4 y 0 –3 less than 1 and is added to the value 1, so the graph is a vertical stretch of followed by a reflection across the x-axis, and then a translation 1 unit up. The domain is {x|x ≥ 0}, and the range is {y|y ≤ 1}. 5 ≈ – 4.2 6 ≈ – 5.2 This graph is the result of a vertical stretch of the graph of followed by a reflection across the x-axis, and then a translation 3 units right. The domain is {x|x ≥ 3}, and the range is {y|y ≤ 0}. SOLUTION: Since f (x) cannot be negative, the minimum point of the graph is where f (x) = 0. Make a table of values. x 0 1 f (x) 5 3 2 1 3 1 4 3 67. SOLUTION: x 0 1 y 1 –1 2 ≈ – 1.8 3 ≈ – 2.5 The domain is all real numbers, and the range is {y | y ≥ 0}. 69. h(x) = |x − 1| SOLUTION: is multiplied by a value The parent function less than 1 and is added to the value 1, so the graph is a vertical stretch of followed by a reflection across the x-axis, and then a translation 1 unit up. The domain is {x|x ≥ 0}, and the range is {y|y ≤ 1}. Determine the domain and range for each function. 68. f (x) = |2x − 5| SOLUTION: Since f (x) cannot be negative, the minimum point of the graph is where f (x) = 0. eSolutions Manual - Powered by Cognero Since h(x) cannot be negative, the minimum point of the graph is where h(x) = 0. Make a table of values. x –4 –2 h(x) 5 3 0 1 2 1 4 3 Page 13 The domain is all real numbers, and the range is {y | y ≥ 0}. 10-2 Simplifying Radical Expressions 69. h(x) = |x − 1| The domain is all real numbers, and the range is {y | y ≥ 0}. 70. SOLUTION: Since h(x) cannot be negative, the minimum point of the graph is where h(x) = 0. Make a table of values. x –4 –2 h(x) 5 3 0 1 2 1 4 3 The domain is all real numbers, and the range is {y | y ≥ 0}. SOLUTION: This is a piecewise-defined function. Make a table of values. Be sure to include the domain values for which the function changes. x -1 0 1 2 3 g(x) -2 -1 0 1 -5 Notice that both functions are linear. The domain is all real numbers, and the range is {y | y ≤ 1}. Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary. 2 71. x − 25 = 0 70. SOLUTION: SOLUTION: For this equation, a = 1, b = 0, and c = –25. This is a piecewise-defined function. Make a table of values. Be sure to include the domain values for which the function changes. x -1 0 1 2 3 g(x) -2 -1 0 1 -5 Notice that both functions are linear. The solutions are 5 and –5. 2 72. r + 25 = 0 eSolutions Manual - Powered by Cognero The domain is all real numbers, and the range is {y | SOLUTION: For this equation, a = 1, b = 0, and c = 25. Page 14 10-2The Simplifying solutions Radical are 5 andExpressions –5. 2 72. r + 25 = 0 SOLUTION: For this equation, a = 1, b = 0, and c = 25. The solution is 5. 2 74. 2r + r − 14 = 0 SOLUTION: For this equation, a = 2, b = 1, and c = –14. There are no real positive square roots of –100. Therefore, the solution to this equation is ø. 2 73. 4w + 100 = 40w SOLUTION: Rewrite the equation in standard form. The solutions are –2.9 and 2.4. 2 75. 5v − 7v = 1 SOLUTION: Rewrite the equation in standard form. For this equation, a = 4, b = –40, and c = 100. For this equation, a = 5, b = –7, and c = –1. The solution is 5. 2 74. 2r + r − 14 = 0 SOLUTION: For this equation, a = 2, b = 1, and c = –14. The solutions are −0.1 and 1.5. 2 76. 11z − z = 3 SOLUTION: Rewrite the equation in standard form. For this equation, a = 11, b = –1, and c = –3. eSolutions Manual - Powered by Cognero Page 15 SOLUTION: 10-2 Simplifying Radical Expressions The solutions are −0.1 and 1.5. 2 76. 11z − z = 3 SOLUTION: Rewrite the equation in standard form. 4 80. 32x − 2y 4 SOLUTION: For this equation, a = 11, b = –1, and c = –3. 2 81. 4t − 27 The solutions are −0.5 and 0.6. Factor each polynomial, if possible. If the polynomial cannot be factored, write prime . 2 77. n − 81 SOLUTION: SOLUTION: In this trinomial, a = 4, b = 0 and c = –27, so m + p is zero and mp is negative. Therefore, m and p must be opposite signs. List the factors of 4(–27) or –108 with a sum of 0. Factors of –108 Sum of 0 1, –108 –107 107 –1, 108 2, –52 –50 50 –2, 52 3, –36 –33 33 –3, 36 4, –27 –23 23 –4, 27 6, –18 –12 12 –6, 18 9, –12 –3 3 –9, 12 2 78. 4 − 9a There are no factors of –108 with a sum of 0, so 4t − 27 is prime. 2 SOLUTION: 3 2 82. x − 3x − 9x + 27 SOLUTION: 5 79. 2x − 98x 3 SOLUTION: 4 80. 32x − 2y 4 SOLUTION: eSolutions Manual - Powered by Cognero 83. POPULATION The country of Latvia has been experiencing a 1.1% annual decrease in population. In 2009, its population was 2,261,294. If the trend continues, predict Latvia’s population in 2019. SOLUTION: Use the equation for exponential decay, with a = 2,261,294, r = 0.011, and t =10. Page 16 86. 88 SOLUTION: 10-2 Simplifying Radical Expressions 83. POPULATION The country of Latvia has been experiencing a 1.1% annual decrease in population. In 2009, its population was 2,261,294. If the trend continues, predict Latvia’s population in 2019. SOLUTION: Use the equation for exponential decay, with a = 2,261,294, r = 0.011, and t =10. 87. 180 SOLUTION: 88. 31 SOLUTION: The number 31 is prime. So, the prime factorization of 31 is 31. 89. 60 Latvia’s population in 2019 will be about 2,024,510 people. 84. TOMATOES There are more than 10,000 varieties of tomatoes. One seed company produces seed packages for 200 varieties of tomatoes. For how many varieties do they not provide seeds? SOLUTION: Let t be the number of varieties of tomatoes for which the seed company does not produce. Since the seed company produces packages for 200 varieties, the total number of varieties can be expressed as: t + 200. Since there are more than 10,000 varieties of tomatoes: t + 200 > 10,000. Solving, we get: t > 10,000 - 200 t > 9,800 SOLUTION: 90. 90 SOLUTION: Write the prime factorization of each number. 85. 24 SOLUTION: 86. 88 SOLUTION: 87. 180 SOLUTION: eSolutions Manual - Powered by Cognero 88. 31 SOLUTION: Page 17 10-3 Operations with Radical Expressions Simplify each expression. 7. 1. SOLUTION: SOLUTION: 2. SOLUTION: 8. SOLUTION: 3. SOLUTION: 9. SOLUTION: 4. SOLUTION: 5. SOLUTION: 10. SOLUTION: 6. SOLUTION: 11. SOLUTION: 7. 12. SOLUTION: eSolutions Manual - Powered by Cognero SOLUTION: Page 1 SOLUTION: SOLUTION: 10-3 Operations with Radical Expressions 16. 12. SOLUTION: SOLUTION: 13. GEOMETRY The area A of a triangle can be found by using the formula , where b represents the base and h is the height. What is the area of the triangle shown? 17. SOLUTION: SOLUTION: 18. SOLUTION: 19. SOLUTION: The area of the triangle is Simplify each expression. . 20. 14. SOLUTION: SOLUTION: 15. SOLUTION: 21. SOLUTION: eSolutions Manual - Powered by Cognero 16. SOLUTION: Page 2 10-3 Operations with Radical Expressions 26. GEOMETRY Find the perimeter and area of a 21. and a length of rectangle with a width of SOLUTION: . SOLUTION: 22. SOLUTION: The perimeter is units. 23. SOLUTION: The area is 12 square units. Simplify each expression. 24. 27. SOLUTION: SOLUTION: 25. SOLUTION: 28. SOLUTION: 26. GEOMETRY Find the perimeter and area of a rectangle with a width of and a length of . SOLUTION: eSolutions Manual - Powered by Cognero Page 3 29. 10-3 Operations with Radical Expressions 32. 28. SOLUTION: SOLUTION: 33. ROLLER COASTERS The velocity v in feet per second of a roller coaster at the bottom of a hill is related to the vertical drop h in feet and the velocity v0 of the coaster at the top of the hill by the formula 29. . SOLUTION: a. What velocity must a coaster have at the top of a 225-foot hill to achieve a velocity of 120 feet per second at the bottom? b. Explain why v0 = v − is not equivalent to the formula given. SOLUTION: a. Let v = 120 and h = 225. 30. SOLUTION: The coaster must have a velocity of 0 feet per second at the top of the hill. 31. SOLUTION: b. Sample answer: In the formula given, we are taking the square root of the difference, not the square root of each term. 34. FINANCIAL LITERACY Tadi invests $225 in a savings account. In two years, Tadi has $232 in his account. You can use the formula to find the average annual interest rate r that the account has earned. The initial investment is v0 and 32. SOLUTION: eSolutions Manual - Powered by Cognero v2 is the amount in two years. What was the average annual interest rate that Tadi’s account earned? SOLUTION: Let v0 = 225 and let v2 = 232. Page 4 second at the top of the hill. b. Sample answer: In the formula given, we are the square of the Expressions difference, not the 10-3taking Operations withroot Radical square root of each term. 34. FINANCIAL LITERACY Tadi invests $225 in a savings account. In two years, Tadi has $232 in his account. You can use the formula The average annual interest rate that Tadi’s account earned was about 1.5%. 35. ELECTRICITY Electricians can calculate the electrical current in amps A by using the formula to find the average annual interest rate r that the account has earned. The initial investment is v0 and v2 is the amount in two years. What was the average annual interest rate that Tadi’s account earned? SOLUTION: Let v0 = 225 and let v2 = 232. , where w is the power in watts and r the resistance in ohms. How much electrical current is running through a microwave oven that has 850 watts of power and 5 ohms of resistance? Write the number of amps in simplest radical form, and then estimate the amount of current to the nearest tenth. SOLUTION: Let w = 850 and let r = 5. The average annual interest rate that Tadi’s account earned was about 1.5%. There are or about 13 amps of electrical current running through the microwave oven. 35. ELECTRICITY Electricians can calculate the electrical current in amps A by using the formula , where w is the power in watts and r the 36. CHALLENGE Determine whether the following statement is true or false . Provide a proof or counterexample to support your answer. x +y > resistance in ohms. How much electrical current is running through a microwave oven that has 850 watts of power and 5 ohms of resistance? Write the number of amps in simplest radical form, and then estimate the amount of current to the nearest tenth. SOLUTION: Let w = 850 and let r = 5. when x > 0 and y > 0 SOLUTION: Because x and y are both positive, each side of the inequality represents a positive number. So, you can prove the statement by squaring each side of the inequality. Because x > 0 and y > 0, the product 2xy must be a positive number.Thus, 2xy > 0 is always true. There are or about 13 amps of electrical current running through the microwave oven. eSolutions Manual - Powered by Cognero 36. CHALLENGE Determine whether the following statement is true or false . Provide a proof or Therefore, for all x > 0 and y > 0. is a true statement Page 5 37. CCSS ARGUMENTS Make a conjecture about are or about 13 amps of electrical 10-3There Operations with Radical Expressions current running through the microwave oven. 36. CHALLENGE Determine whether the following statement is true or false . Provide a proof or counterexample to support your answer. when x > 0 and y > 0 x +y > SOLUTION: Because x and y are both positive, each side of the inequality represents a positive number. So, you can prove the statement by squaring each side of the inequality. Therefore, for all x > 0 and y > 0. is a true statement 37. CCSS ARGUMENTS Make a conjecture about the sum of a rational number and an irrational number. Is the sum rational or irrational? Is the product of a nonzero rational number and an irrational number rational or irrational? Explain your reasoning. SOLUTION: Examine the sum of several pairs of rational and irrational numbers: is in lowest terms, and is irrational. is in lowest terms and is irrational. Because x > 0 and y > 0, the product 2xy must be a positive number.Thus, 2xy > 0 is always true. Therefore, for all x > 0 and y > 0. is a true statement is irrational. Examine the product of several pairs of non-zero rational and irrational numbers: is irrational is irrational 37. CCSS ARGUMENTS Make a conjecture about the sum of a rational number and an irrational number. Is the sum rational or irrational? Is the product of a nonzero rational number and an irrational number rational or irrational? Explain your reasoning. SOLUTION: Examine the sum of several pairs of rational and irrational numbers: is in lowest terms, and is irrational. is in lowest terms and is irrational. is irrational. From the above examples, we should come up with the conjecture that the sum of a rational number and an irrational number is irrational, and the product of a rational number and an irrational number is irrational. 38. OPEN ENDED Write an equation that shows a sum of two radicals with different radicands. Explain how you could combine these terms. SOLUTION: Sample answer: is irrational. Examine the product of several pairs of non-zero rational and irrational numbers: is irrational When you simplify eSolutions Manual - Powered by Cognero is irrational is irrational. , you get . When you simplify , you get . Because these two numbers have the same radicand, you can add them. Page 6 39. WRITING IN MATH Describe step by step how to multiply two radical expressions, each with two the conjecture that the sum of a rational number and an irrational number is irrational, and the product of a rational number and an irrational number is irrational. 10-3 Operations with Radical Expressions 38. OPEN ENDED Write an equation that shows a sum of two radicals with different radicands. Explain how you could combine these terms. In 6 years, the population of the town will be 14,500. 41. GEOMETRY Which expression represents the sum of the lengths of the 12 edges on this rectangular solid? SOLUTION: Sample answer: A 2(a + b + c) B 3(a + b + c) When you simplify , you get . When you simplify , you get . Because these two numbers have the same radicand, you can add them. C 4(a + b + c) D 12(a + b + c) 39. WRITING IN MATH Describe step by step how to multiply two radical expressions, each with two terms. Write an example to demonstrate your description. SOLUTION: For example, you can use the FOIL method. You multiply the first terms within the parentheses. Then you multiply the outer terms within the parentheses. Then you would multiply the inner terms within the parentheses. And, then you would multiply the last terms within each parentheses. Combine any like terms and simplify any radicals. For example: 40. SHORT RESPONSE The population of a town is 13,000 and is increasing by about 250 people per year. This can be represented by the equation p = 13,000 + 250y, where y is the number of years from now and p represents the population. In how many years will the population of the town be 14,500? SOLUTION: Let p = 14,500. SOLUTION: There are 4 edges that have a length of a, 4 edges that have a length of b, and 4 edges that have a length of c. So, the expression 4a + 4b + 4c or 4(a + b + c) represents the sum of the lengths of the 12 edges of the rectangular solid. Choice C is the correct answer. 42. Evaluate F 4; 4 G 4; 2 H 2; 4 J 2; 2 and for n = 25. SOLUTION: Substitute 25 for n in both expressions. Choice G is correct. 43. The current I in a simple electrical circuit is given by the formula In 6 years, the population of the town will be 14,500. Which expression 41. GEOMETRY eSolutions Manual - Powered by Cognero represents the sum of the lengths of the 12 edges on this rectangular solid? , where V is the voltage and R is the resistance of the circuit. If the voltage remains unchanged, what effect will doubling the resistance of the circuit have on the current? A The current will remain the same. Page 7 10-3 Operations with Radical Expressions Choice G is correct. 43. The current I in a simple electrical circuit is given by the formula 45. , where V is the voltage and R is SOLUTION: the resistance of the circuit. If the voltage remains unchanged, what effect will doubling the resistance of the circuit have on the current? A The current will remain the same. B The current will double its previous value. 46. C The current will be half its previous value. SOLUTION: D The current will be two units more than its previous value. SOLUTION: The current and the resistance have an inverse relationship. If the resistance is double and the voltage remains the same, the current will be half of its previous value. 47. Consider an example with the resistance is 4 and voltage is 100. Find I when R doubles. SOLUTION: 48. SOLUTION: Choice C is the correct answer. 49. SOLUTION: Simplify. 44. SOLUTION: Graph each function. Compare to the parent graph. State the domain and range. 50. eSolutions Manual - Powered by Cognero 45. SOLUTION: SOLUTION: x 0 y 0 0.5 ≈ 1.4 1 2 2 ≈ 2.8 Page 8 and a reflection across the x-axis. Another way to identify the stretch is to notice that the yvalues in the table are –3 times the corresponding yvalues for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}. SOLUTION: 10-3 Operations with Radical Expressions Graph each function. Compare to the parent graph. State the domain and range. 52. SOLUTION: x –1 –0.5 y 0 ≈ 0.7 50. SOLUTION: x 0 y 0 0.5 ≈ 1.4 1 2 0 1 1 ≈ 1.4 2 ≈ 1.7 3 ≈ 2 2 ≈ 2.8 The value 1 is being added to the square root of the parent function , so the graph is translated 1 is multiplied by a value The parent function greater than 1, so the graph is a vertical stretch of . Another way to identify the stretch is to notice that the y-values in the table are 2 times the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}. unit left from the parent graph . Another way to identify the translation is to note that the x-values in the table are 1 less than the corresponding xvalues for the parent function. The domain is {x|x ≥ –1}, and the range is {y|y ≥ 0}. 53. 51. SOLUTION: x 0 y 0 0.5 ≈ –2.1 1 –3 SOLUTION: x 4 4.5 y 0 ≈ 0.7 2 ≈ –4.2 5 1 6 ≈ 1.4 7 ≈ 1.7 8 2 The value 4 is being subtracted from the square root of the parent function , so the graph is translated 4 units right from the parent graph . Another way to identify the translation is to note that the x-values in the table are 4 more than the corresponding x-values for the parent function. The domain is {x|x ≥ 4}, and the range is {y|y ≥ 0}. The parent function is multiplied by a value less than 1, so the graph is a vertical stretch of and a reflection across the x-axis. Another way to identify the stretch is to notice that the yvalues in the table are –3 times the corresponding yvalues for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}. 54. 52. SOLUTION: eSolutions Manual - Powered by Cognero x 0 1 –1 –0.5 y 0 1 ≈ 0.7 ≈ 1.4 2 ≈ 1.7 3 ≈ 2 SOLUTION: x 0 0.5 y 3 ≈ 3.7 1 4 2 ≈ 4.4 3 ≈ 4.7 4 5 Page 9 translated 4 units right from the parent graph . Another way to identify the translation is to note that the x-values in the table are 4 more than the 10-3corresponding Operations with Radical Expressions x-values for the parent function. The domain is {x|x ≥ 4}, and the range is {y|y ≥ 0}. units from the parent graph . Another way to identify the translation is to note that the y-values in the table are 2 less than the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ –2}. Factor each trinomial. 2 56. x + 12x + 27 54. SOLUTION: x 0 0.5 y 3 ≈ 3.7 1 4 2 ≈ 4.4 3 ≈ 4.7 4 5 SOLUTION: 2 57. y + 13y + 30 SOLUTION: 2 The value 3 is being added to the parent function , so the graph is translated up 3 units from the parent graph . Another way to identify the translation is to note that the y-values in the table are 3 greater than the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 3}. 58. p − 17p + 72 SOLUTION: 2 59. x + 6x – 7 SOLUTION: 55. SOLUTION: x 0 0.5 y –2 ≈ – 1.3 1 –1 2 ≈ – 0.6 3 ≈ – 0.3 4 0 2 60. y − y − 42 SOLUTION: 2 61. −72 + 6w + w SOLUTION: The value 2 is being subtracted from the parent function , so the graph is translated down 2 units from the parent graph . Another way to identify the translation is to note that the y-values in the table are 2 less than the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ –2}. 62. FINANCIAL LITERACY Determine the value of an investment if $400 is invested at an interest rate of 7.25% compounded quarterly for 7 years. SOLUTION: Use the formula for calculating compound interest. Factor each trinomial. 2 56. x + 12x + 27 eSolutions Manual - Powered by Cognero SOLUTION: Page 10 SOLUTION: 10-3 Operations with Radical Expressions 62. FINANCIAL LITERACY Determine the value of an investment if $400 is invested at an interest rate of 7.25% compounded quarterly for 7 years. SOLUTION: Use the formula for calculating compound interest. 67. SOLUTION: 68. 3.6t + 6 − 2.5t = 8 The value of the investment after 7 years is about $661.44. SOLUTION: Solve each equation. Round each solution to the nearest tenth, if necessary. 63. −4c − 1.2 = 0.8 SOLUTION: 64. −2.6q − 33.7 = 84.1 SOLUTION: 65. 0.3m + 4 = 9.6 SOLUTION: 66. SOLUTION: 67. eSolutions Manual - Powered by Cognero SOLUTION: Page 11 10-4 Radical Equations 1. GEOMETRY The surface area of a basketball is x square inches. What is the radius of the basketball if the formula for the surface area of a sphere is SA = 3. SOLUTION: 2 4πr ? SOLUTION: The surface area is x square inches, so substitute x 2 for SA in the formula SA = 4πr , then solve for r. Check: 4. SOLUTION: So, the radius of the basketball is inches. Solve each equation. Check your solution. 2. SOLUTION: Check: Check: 5. SOLUTION: 3. SOLUTION: eSolutions Manual - Powered by Cognero Check: Page 1 Because –1 does not satisfy the original equation, 3 is the only solution. 10-4 Radical Equations 5. 7. SOLUTION: SOLUTION: Check: Check: Because 3 does not satisfy the original equation, 10 is the only solution. Because 3 does not satisfy the original equation, 6 is the only solution. 8. EXERCISE Suppose the function 6. SOLUTION: Check: , where S represents speed in meters per second and is the leg length of a person in meters, can approximate the maximum speed that a person can run. a. What is the maximum running speed of a person with a leg length of 1.1 meters to the nearest tenth of a meter? b. What is the leg length of a person with a running speed of 6.7 meters per second to the nearest tenth of a meter? c. As leg length increases, does maximum speed increase or decrease? Explain. SOLUTION: a. Substitute 1.1 for in the equation . Because –1 does not satisfy the original equation, 3 is the only solution. 7. SOLUTION: The maximum running speed is about 8.2 meters per second. b. Substitute 6.7 for S in the equation eSolutions Manual - Powered by Cognero Check: . Page 2 0.7 6.5 1.1 8.2 1.5 9.5 As the person’s leg length increases, the speed increases. 10-4Because Radical3Equations does not satisfy the original equation, 6 is the only solution. 8. EXERCISE Suppose the function Solve each equation. Check your solution. , where S represents speed in meters per second and is the leg length of a person in meters, can approximate the maximum speed that a person can run. a. What is the maximum running speed of a person with a leg length of 1.1 meters to the nearest tenth of a meter? b. What is the leg length of a person with a running speed of 6.7 meters per second to the nearest tenth of a meter? c. As leg length increases, does maximum speed increase or decrease? Explain. SOLUTION: a. Substitute 1.1 for in the equation 9. SOLUTION: Check: . 10. SOLUTION: The maximum running speed is about 8.2 meters per second. b. Substitute 6.7 for S in the equation . Check: 11. SOLUTION: The leg length is about 0.7 meters. c. 0.7 6.5 1.1 8.2 1.5 9.5 As the person’s leg length increases, the speed increases. Check: Solve each equation. Check your solution. eSolutions Manual - Powered by Cognero Page 3 9. 12. 10-4 Radical Equations 11. 14. SOLUTION: SOLUTION: Check: Check: 12. SOLUTION: 15. SOLUTION: Check: Check: 13. SOLUTION: Because –4 does not satisfy the original equation, 3 is the only solution. Check: 16. SOLUTION: Check: eSolutions Manual - Powered by Cognero 14. SOLUTION: Page 4 10-4Because Radical –4 Equations does not satisfy the original equation, 3 is the only solution. 16. Because 1 does not satisfy the original equation, 6 is the only solution. 18. SOLUTION: SOLUTION: Check: Check: Because –2 does not satisfy the original equation, 3 is the only solution. 17. Because –4 does not satisfy the original equation, 0 is the only solution. SOLUTION: 19. SOLUTION: Check: Check: Because 1 does not satisfy the original equation, 6 is the only solution. 18. SOLUTION: The solution is 7. 20. SOLUTION: Check: eSolutions Manual - Powered by Cognero Page 5 10-4 Radical Equations The solution is 7. The solution is 47. 21. RIDES The amount of time t, in seconds, that it takes a simple pendulum to complete a full swing is 20. SOLUTION: called the period . It is given by , where is the length of the pendulum, in feet. a. The Giant Swing completes a period in about 8 seconds. About how long is the pendulum’s arm? Round to the nearest foot. b. Does increasing the length of the pendulum increase or decrease the period? Explain. Check: SOLUTION: a. Substitute 8 for t in the equation . The solution is 47. 21. RIDES The amount of time t, in seconds, that it takes a simple pendulum to complete a full swing is called the period . It is given by The length of the pendulum’s arm is about 52 feet. b. , where is the length of the pendulum, in feet. a. The Giant Swing completes a period in about 8 seconds. About how long is the pendulum’s arm? Round to the nearest foot. b. Does increasing the length of the pendulum increase or decrease the period? Explain. 52 8.00 55 8.24 60 8.60 As the length of the pendulum increases, the period also increases. SOLUTION: a. Substitute 8 for t in the equation Solve each equation. Check your solution. . 22. SOLUTION: The length of the pendulum’s arm is about 52 feet. b. 52 Use the quadratic equation to solve, with a = 1, b = – 8.00 eSolutions - Powered by Cognero 55 Manual8.24 60 8.60 As the length of the pendulum increases, the period Page 6 52 8.00 55 8.24 60 8.60 10-4As Radical Equations the length of the pendulum increases, the period also increases. This equation has no solution. Solve each equation. Check your solution. 24. 22. SOLUTION: SOLUTION: Use the quadratic equation to solve, with a = 1, b = – Check: Since you cannot take the square root of a negative n equation has no real solution. 23. SOLUTION: Check: The solution is . 25. SOLUTION: This equation has no solution. 24. SOLUTION: eSolutions Manual - Powered by Cognero Page 7 Check: Because –11 does not satisfy the original equation, 11 is the only solution. 10-4The Radical Equations solution is . 27. 25. SOLUTION: SOLUTION: Check: Check: Because –3 does not satisfy the original equation, 3 is the only solution. 28. CCSS REASONING The formula for the slant height c of a cone is , where h is the height of the cone and r is the radius of its base. Find the height of the cone if the slant height is 4 and the radius is 2. Round to the nearest tenth. The solution is 235.2. 26. SOLUTION: SOLUTION: Substitute 4 for c and 2 for r in the equation . Check: Because –11 does not satisfy the original equation, 11 is the only solution. 27. SOLUTION: eSolutions Manual - Powered by Cognero The height of the cone is or ≈ 3.5. Page 8 29. MULTIPLE REPRESENTATIONS Consider a. GRAPHICAL Clear the Y= list. Enter the left s 10-4 Radical Equations The height of the cone is or ≈ 3.5. Use the quadratic equation to solve, with a = 1, b = – 29. MULTIPLE REPRESENTATIONS Consider a. GRAPHICAL Clear the Y= list. Enter the left s x − 7. Press GRAPH. b. GRAPHICAL Sketch what is shown on the scr c. ANALYTICAL Use the intersect feature on t d. ANALYTICAL Solve the radical equation alge SOLUTION: a. Check: Use the WINDOW menu to adjust the viewing wi Because 5.17 does not satisfy the original equation, 1 calculator. [-10, 20] scl: 1 by [-10, 10] scl: 1 b. See students’ work. c. 30. PACKAGING A cylindrical container of chocolate drink mix has a volume of 162 cubic inches. The radius r of the container can be found by using the formula , where V is the volume of the container and h is the height. a. If the radius is 2.5 inches, find the height of the container. Round your answer to the nearest hundredth. b. If the height of the container is 10 inches, find the radius. Round to the nearest hundredth. [-10, 20] scl: 1 by [-10, 10] scl: 1 [-10, 20] scl: 1 by [ d. SOLUTION: a. Substitute 162 for V and 2.5 for r in the equation . Use the quadratic equation to solve, with a = 1, b = – eSolutions Manual - Powered by Cognero Page 9 10-4Because Radical5.17 Equations does not satisfy the original equation, 1 calculator. 30. PACKAGING A cylindrical container of chocolate drink mix has a volume of 162 cubic inches. The radius r of the container can be found by using the formula The radius of the container is about 2.27 inches. 31. CCSS CRITIQUE Jada and Fina solved . Is either of them correct? Explain. , where V is the volume of the container and h is the height. a. If the radius is 2.5 inches, find the height of the container. Round your answer to the nearest hundredth. b. If the height of the container is 10 inches, find the radius. Round to the nearest hundredth. SOLUTION: SOLUTION: a. Substitute 162 for V and 2.5 for r in the equation . Jada is correct. Fina had the wrong sign for 2b in the fourth step. 32. REASONING Which equation has the same solution set as ? Explain. A. B. 4 = x + 2 C. SOLUTION: The height of the container is about 8.25 inches. b. Substitute 162 for V and 10 for h in the equation . The solution to the equation is 2. In choice A, . Choice B is the result of squaring both sides of the equation, and 4 = 2 + 2. The radius of the container is about 2.27 inches. 31. CCSS CRITIQUE Jada and Fina solved . Is either of them correct? Explain. eSolutions Manual - Powered by Cognero In choice C, . So, the correct choice is B. 33. REASONING Explain how solving different from solving SOLUTION: . is Page 10 equation, and 4 = 2 + 2. In choice C, . 10-4 Radical Equations So, the correct choice is B. The solution is 1. 33. REASONING Explain how solving different from solving is . SOLUTION: In the first equation, you have to isolate the radical first by subtracting 1 from each side. Then square each side to find the value of x. In the second equation, the radical is already isolated, so square each side to start. Then subtract 1 from each side to solve for x. Solve 35. REASONING Is the following equation sometimes, always or never true? Explain. SOLUTION: Sometimes; the equation is true for x ≥ 2, but false for x < 2, because if x is less than 2 then the right side of the equation will be negative, and a square root can never be negative. . When x = 3, Solve . when x = 2, 34. OPEN ENDED Write a radical equation with a variable on each side. Then solve the equation. SOLUTION: Sample answer: When x = 1. 36. CHALLENGE Solve . SOLUTION: Check: Check: The solution is 1. 35. REASONING Is the following equation sometimes, eSolutions Manual Powered by Cognero always or -never true? Explain. Page 11 10-4 Radical Equations When x = 1. 36. CHALLENGE Solve . SOLUTION: Check: 37. WRITING IN MATH Write some general rules about how to solve radical equations. Demonstrate your rules by solving a radical equation. SOLUTION: Start by using addition/subtraction to isolate the term containing the radical on one side of the equation. Then multiply/divide each side by a number to change the coefficient of the radical to +1.Next, square each side of the equation to eliminate the radical. Solve the resulting equation using the most appropriate method. Last, check the answers found in the original equation to remove any extraneous solutions. 37. WRITING IN MATH Write some general rules about how to solve radical equations. Demonstrate your rules by solving a radical equation. SOLUTION: Start by using addition/subtraction to isolate the term containing the radical on one side of the equation. Then multiply/divide each side by a number to change the coefficient of the radical to +1.Next, square each side of the equation to eliminate the radical. Solve the resulting equation using the most appropriate method. Last, check the answers found in the original equation to remove any extraneous solutions. Check: There are no extraneous solutions. Therefore, the solution is eSolutions Manual - Powered by Cognero Check: . 38. SHORT RESPONSE Zach needs to drill a hole at Page 12 A, B, C, D, and E on circle P. There are no extraneous solutions. Therefore, the 10-4solution RadicalisEquations . 38. SHORT RESPONSE Zach needs to drill a hole at A, B, C, D, and E on circle P. The measure of ∠CPD is 62.5°. 39. Which expression is undefined when w = 3? A B C If Zack drills holes so that m APE = 110º and the other four angles are congruent, what is m CPD? SOLUTION: The measure of the central angles in a circle sum to 360, so . Let x = the measure of each of the other 4 angles. D SOLUTION: For A; For B; The measure of ∠CPD is 62.5°. 39. Which expression is undefined when w = 3? A For C; B C D For D; SOLUTION: For A; The expression in C is undefined when w = 3. For B; 40. What is the slope of a line that is parallel to the line? For C; F −3 eSolutions Manual - Powered by Cognero G Page 13 The slope of the line on the graph is parallel to the line on the graph has the same slope. So the correct choice is H. 10-4 Radical Equations The expression in C is undefined when w = 3. 40. What is the slope of a line that is parallel to the line? . A line that is 41. What are the solutions of A 1, 6 B −1, −6 C 1 D 6 ? SOLUTION: F −3 G H J 3 Check: SOLUTION: First find the slope of the line on the graph. Use the two points (0, 2) and (3, 3). Because 1 does not satisfy the original equation, 6 is the only solution. So, the correct choice is D. The slope of the line on the graph is . A line that is 42. ELECTRICITY The voltage V required for a circuit is given by , where P is the power in watts and R is the resistance in ohms. How many more volts are needed to light a 100-watt light bulb than a 75-watt light bulb if the resistance of both is 110 ohms? parallel to the line on the graph has the same slope. So the correct choice is H. 41. What are the solutions of A 1, 6 B −1, −6 C 1 D 6 ? SOLUTION: First find the number of volts needed to light a 100watt light bulb. Substitute 100 for P and 110 for R in the equation . SOLUTION: Next find the number of volts needed to light a 75watt light bulb. Substitute 75 for P and 110 for R in the equation . Check: So, 105 – 91, or about 14 more volts are needed to light a 100-watt light bulb. Simplify each expression. eSolutions Manual - Powered by Cognero Because 1 does not satisfy the original equation, 6 is 43. Page 14 105 – Equations 91, or about 14 more volts are needed to 10-4So, Radical light a 100-watt light bulb. Simplify each expression. 47. 43. SOLUTION: SOLUTION: 44. SOLUTION: 48. SOLUTION: 45. SOLUTION: 49. PHYSICAL SCIENCE A projectile is shot straight up from ground level. Its height h, in feet, after t 46. 2 SOLUTION: seconds is given by h = 96t − 16t . Find the value(s) of t when h is 96 feet. SOLUTION: Use the quadratic equation to solve, with a = –16, b = 96, and c = –96. 47. eSolutions Manual - Powered by Cognero SOLUTION: Page 15 Therefore, the projectile will be at 96 feet on the way up at 1.3 seconds, and will be at 96 feet on the way down at 4.7 seconds. 10-4 Radical Equations 49. PHYSICAL SCIENCE A projectile is shot straight up from ground level. Its height h, in feet, after t 2 seconds is given by h = 96t − 16t . Find the value(s) of t when h is 96 feet. SOLUTION: Use the quadratic equation to solve, with a = –16, b = 96, and c = –96. Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime . 2 50. 2x + 7x + 5 SOLUTION: In this trinomial, a = 2, b = 7 and c = 5, so m + p is positive and mp is positive. Therefore, m and p must both be positive. List the factors of 2(5) or 10. Identify the factors with a sum of 7. Factors of 10 Sum 1, 10 11 2, 5 7 The correct factors are 2 and 5. 2 So, 2x + 7x + 5= (2x + 5) (x + 1). Therefore, the projectile will be at 96 feet on the way up at 1.3 seconds, and will be at 96 feet on the way down at 4.7 seconds. Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime . 2 50. 2x + 7x + 5 SOLUTION: In this trinomial, a = 2, b = 7 and c = 5, so m + p is positive and mp is positive. Therefore, m and p must both be positive. List the factors of 2(5) or 10. Identify the factors with a sum of 7. Factors of 10 Sum 1, 10 11 2, 5 7 The correct factors are 2 and 5. 2 So, 2x + 7x + 5= (2x + 5) (x + 1). eSolutions Manual - Powered by Cognero 2 51. 6p + 5p − 6 SOLUTION: 2 51. 6p + 5p − 6 SOLUTION: In this trinomial, a = 6, b = 5 and c = –6, so m + p is positive and mp is negative. Therefore, m and p must be opposite signs. List the factors of 6(–6) or –36 with a sum of 5. Factors of –36 Sum 1, –36 –35 35 –1, 36 2, –18 –16 16 –2, 18 3, –12 –9 9 –3, 12 4, –9 –5 5 –4, 9 The correct factors are –4 and 9. 2 So, 6p + 5p − 6= (2p + 3) (3p − 2). 2 52. 5d + 6d − 8 SOLUTION: Page 16 2 10-4So, Radical 2x + Equations 7x + 5= (2x + 5) (x + 1). 2 2 So, 6p + 5p − 6= (2p + 3) (3p − 2). 2 51. 6p + 5p − 6 52. 5d + 6d − 8 SOLUTION: In this trinomial, a = 6, b = 5 and c = –6, so m + p is positive and mp is negative. Therefore, m and p must be opposite signs. List the factors of 6(–6) or –36 with a sum of 5. Factors of –36 Sum 1, –36 –35 35 –1, 36 2, –18 –16 16 –2, 18 3, –12 –9 9 –3, 12 4, –9 –5 5 –4, 9 The correct factors are –4 and 9. 2 So, 6p + 5p − 6= (2p + 3) (3p − 2). SOLUTION: In this trinomial, a = 5, b = 6 and c = –8, so m + p is positive and mp is negative. Therefore, m and p must be opposite signs. List the factors of 5(–8) or –40 with a sum of 6. Factors of –40 Sum 1, –40 –39 39 –1, 40 2, –20 –18 18 –2, 20 4, –10 –6 6 –4, 10 5, –8 –3 3 –5, 8 The correct factors are –4 and 10. 2 So, 5d + 6d − 8= (5d − 4) (d + 2). 2 2 53. 8k − 19k + 9 52. 5d + 6d − 8 SOLUTION: In this trinomial, a = 5, b = 6 and c = –8, so m + p is positive and mp is negative. Therefore, m and p must be opposite signs. List the factors of 5(–8) or –40 with a sum of 6. Factors of –40 Sum 1, –40 –39 39 –1, 40 2, –20 –18 18 –2, 20 4, –10 –6 6 –4, 10 5, –8 –3 3 –5, 8 The correct factors are –4 and 10. SOLUTION: In this trinomial, a = 8, b = –19 and c = 9, so m + p is negative and mp is positive. Therefore, m and p must both be negative. List the negative factors of 8(9) or 72 with a sum of –19. Factors of 72 Sum –1, –72 –73 –2, –36 –38 –3, –24 –27 –4, –18 –22 –6, –12 –18 –8, –9 –17 There are no negative factors of 72 with a sum of – 19. 2 So, 5d + 6d − 8 is prime. 2 54. 9g − 12g + 4 2 eSolutions Manual - Powered by Cognero So, 5d + 6d − 8= (5d − 4) (d 2 53. 8k − 19k + 9 + 2). SOLUTION: In this trinomial, a = 9, b = –12 and c = 4, so m + p is negative and mp is positive. Therefore, m and p must both be negative. List the negative factors of 9(4) or 36 with a sum of –12. Page 17 Factors of 36 Sum –1, –36 –37 –2, –18 –20 –6, –12 –18 –8, –9 –17 There are no negative factors of 72 with a sum of – 10-419. Radical Equations 2 2 So, 5d + 6d − 8 is prime. So, 2a − 9a − 18= (2a + 3)(a − 6). 2 Determine whether each expression is a monomial. Write yes or no. Explain. 54. 9g − 12g + 4 SOLUTION: In this trinomial, a = 9, b = –12 and c = 4, so m + p is negative and mp is positive. Therefore, m and p must both be negative. List the negative factors of 9(4) or 36 with a sum of –12. Factors of 36 Sum –1, –36 –37 –2, –18 –20 –3, –12 –15 –4, –9 –13 –6, –6 –12 The correct factors are –6 and –6. 56. 12 SOLUTION: Yes; 12 is a real number and therefore a monomial. 3 57. 4x SOLUTION: 3 Yes; 4x is the product of a number and three variables. 58. a − 2b SOLUTION: No; a – 2b shows subtraction, not multiplication alone of numbers and variables. 59. 4n + 5p SOLUTION: No; 4n + 5p shows addition, not multiplication alone of numbers and variables. 2 So, 9g − 12g + 4= (3g − 2)(3g − 2). 2 55. 2a − 9a − 18 SOLUTION: In this trinomial, a = 2, b = –9 and c = –18, so m + p is negative and mp is negative. Therefore, m and p must be opposite signs. List the factors of 2(–18) or –36 with a sum of –9. Factors of –36 Sum 1, –36 –35 35 –1, 36 2, –18 –16 16 –2, 18 3, –12 –9 9 –3, 12 4, –9 –5 5 –4, 9 The correct factors are 3 and –12. 60. SOLUTION: No; has a variable in the denominator. 61. SOLUTION: Yes; is the product of a number, , and several variables. Simplify. 2 62. 9 SOLUTION: 2 So, 2a − 9a − 18= (2a + 3)(a − 6). Determine whether each expression is a monomial. Write yes or no. Explain. eSolutions Manual - Powered by Cognero 56. 12 6 63. 10 SOLUTION: Page 18 SOLUTION: 10-4 Radical Equations 6 63. 10 SOLUTION: 5 64. 4 SOLUTION: 65. (8v) 2 SOLUTION: 66. SOLUTION: 2 3 67. (10y ) SOLUTION: eSolutions Manual - Powered by Cognero Page 19 10-5 The Pythagorean Theorem Find each missing length. If necessary, round to the nearest hundredth. 4. 1. SOLUTION: Use the Pythagorean Theorem, substituting 3 for a and 4 for b. SOLUTION: Use the Pythagorean Theorem, substituting 8 for a and 12 for b. 5. BASEBALL A baseball diamond is a square. The distance between consecutive bases is 90 feet. 2. SOLUTION: Use the Pythagorean Theorem, substituting 4 for a and 21 for c. 3. SOLUTION: Use the Pythagorean Theorem, substituting 6 for b and 19 for c. a. How far does a catcher have to throw the ball from home plate to second base? b. How far does a third baseman throw the ball to the first baseman from a point in the baseline 15 feet from third to second base? c. A base runner going from first to second base is 100 feet from home plate. How far is the runner from second base? SOLUTION: a. Use the Pythagorean Theorem, substituting 90 for a and 90 for b. The catcher has to throw the ball about 127 ft from home plate to second base. 4. SOLUTION: eSolutions Manual - Powered by Cognero Use the Pythagorean Theorem, substituting 8 for a and 12 for b. b. The diagram below illustrates the throw made by the third baseman. Page 1 has to throw the ball about 127 ft from 10-5The Thecatcher Pythagorean Theorem home plate to second base. b. The diagram below illustrates the throw made by the third baseman. So, the base runner is about 44 feet from first base to second base. Therefore, the distance the base runner is from second is about 90 – 44 or 46 feet. Determine whether each set of measures can be the lengths of the sides of a right triangle. 6. 8, 12, 16 SOLUTION: Since the measure of the longest side is 16, let c = 2 16, a = 8, and b = 12. Then determine whether c = 2 2 a +b . The length of a is 90 – 15 or 75 feet. Find the length of the throw c by using the Pythagorean Theorem, substituting 75 for a and 90 for b. 2 2 2 No, because c ≠ a + b , a triangle with side lengths 8, 12, and 16 is not a right triangle. 7. 28, 45, 53 The third baseman has to throw the ball about 117 feet to the first baseman. SOLUTION: Since the measure of the longest side is 53, let c = 2 53, a = 28, and b = 45. Then determine whether c = 2 2 a +b . c. The diagram below illustrates the position of the base runner. 2 2 2 Yes, because c = a + b , a triangle with side lengths 28, 45, and 53 is a right triangle. 8. 7, 24, 25 Use the Pythagorean Theorem to find the length of side a by substituting 90 for b and 100 for c. SOLUTION: Since the measure of the longest side is 25, let c = 2 25, a = 7, and b = 24. Then determine whether c = 2 2 a +b . 2 So, the base runner is about 44 feet from first base to second base. Therefore, the distance the base runner is from second is about 90 – 44 or 46 feet. eSolutions Manual - Powered by Cognero Determine whether each set of measures can be the lengths of the sides of a right triangle. 6. 8, 12, 16 2 2 Yes, because c = a + b , a triangle with side lengths 7, 24, and 25 is a right triangle. 9. 15, 25, 45 SOLUTION: Page 2 Since the measure of the longest side is 45, let c = 2 45, a = 15, and b = 25. Then determine whether c = 2 2 2 10-5Yes, The because Pythagorean c = a Theorem + b , a triangle with side lengths 7, 24, and 25 is a right triangle. 9. 15, 25, 45 SOLUTION: Since the measure of the longest side is 45, let c = 2 45, a = 15, and b = 25. Then determine whether c = 2 12. SOLUTION: 2 a +b . Use the Pythagorean Theorem, substituting a and 20 for b. 2 2 for 2 No, because c ≠ a + b , a triangle with side lengths 15, 25, and 45 is not a right triangle. Find each missing length. If necessary, round to the nearest hundredth. 13. SOLUTION: Use the Pythagorean Theorem, substituting 9 for b and 31 for c. 10. SOLUTION: Use the Pythagorean Theorem, substituting 6 for a and 14 for b. 14. 11. SOLUTION: Use the Pythagorean Theorem, substituting 16 for a and 26 for c. SOLUTION: Use the Pythagorean Theorem, substituting 2 for a and 12 for c. 15. SOLUTION: 12. eSolutions Manual - Powered by Cognero Use the Pythagorean Theorem, substituting SOLUTION: Use the Pythagorean Theorem, substituting for and for b. Page 3 for a 10-5 The Pythagorean Theorem 15. 18. SOLUTION: Use the Pythagorean Theorem, substituting and for a SOLUTION: Use the Pythagorean Theorem, substituting for b. and 16. SOLUTION: Use the Pythagorean Theorem, substituting 7 for b and 25 for c. 17. SOLUTION: Use the Pythagorean Theorem, substituting 5 for a and for c. for b for c. 19. TELEVISION Larry is buying an entertainment stand for his television. The diagonal of his television is 42 inches. The space for the television measures 30 inches by 36 inches. Will Larry’s television fit? Explain. SOLUTION: Use the Pythagorean Theorem, substituting 30 for a and 36 for b. Yes; sample answer: The diagonal of the space in the TV stand is about 46.9 inches, which is greater than the diagonal of the television, so Larry’s TV will fit. Determine whether each set of measures can be the lengths of the sides of a right triangle. Then determine whether they form a Pythagorean triple. 20. 9, 40, 41 SOLUTION: Since the measure of the longest side is 41, let c = 2 41, a = 9, and b = 40. Then determine whether c = 2 2 a +b . 18. eSolutions Manual - Powered by Cognero SOLUTION: Use the Pythagorean Theorem, substituting Page 4 for b 3, , and is not a right triangle. No, because a Pythagorean triple is a group of three Yes; sample answer: The diagonal of the space in the TV stand is about 46.9 inches, which is greater 10-5than Thethe Pythagorean Theorem diagonal of the television, so Larry’s TV will fit. Determine whether each set of measures can be the lengths of the sides of a right triangle. Then determine whether they form a Pythagorean triple. 20. 9, 40, 41 2 2 2 whole numbers that satisfy the equation c = a + b , where c is the greatest number. 22. SOLUTION: Since the measure of the longest side is 12, let c = 2 12, a = 4, and b = . Then determine whether c 2 SOLUTION: Since the measure of the longest side is 41, let c = 2 =a +b . 2 41, a = 9, and b = 40. Then determine whether c = 2 2 a +b . 2 2 2 2 2 No, because c ≠ a + b , a triangle with side lengths 4, , and 12 is not a right triangle. No, because a Pythagorean triple is a group of three 2 Yes, because c = a + b , a triangle with side lengths 9, 40, and 41 is a right triangle. Yes, because a Pythagorean triple is a group of three 2 2 2 whole numbers that satisfy the equation c = a + b , where c is the greatest number. 2 2 2 whole numbers that satisfy the equation c = a + b , where c is the greatest number. 23. SOLUTION: Since the measure of the longest side is 14, let c = 2 14, a = , and b = 7. Then determine whether c = 21. SOLUTION: 2 2 a +b . Since the measure of the longest side is , let c = , a = 3, and b = . Then determine whether 2 2 2 c =a +b . 2 2 2 No, because c ≠ a + b , a triangle with side lengths , 7, and 14 is not a right triangle. No, because a Pythagorean triple is a group of three 2 2 2 No, because c ≠ a + b , a triangle with side lengths 3, , and is not a right triangle. No, because a Pythagorean triple is a group of three 2 2 2 whole numbers that satisfy the equation c = a + b , where c is the greatest number. 22. 2 2 2 24. 8, 31.5, 32.5 SOLUTION: Since the measure of the longest side is 32.5, let c = 2 32.5, a = 8, and b = 31.5. Then determine whether c 2 SOLUTION: Since the measure of the longest side is 12, let c = 2 12, a = 4, and b = . Then determine whether c 2 whole numbers that satisfy the equation c = a + b , where c is the greatest number. 2 =a +b . 2 =a +b . eSolutions Manual - Powered by Cognero 2 2 2 Page 5 Yes, because c = a + b , a triangle with side lengths 8, 31.5, and 32.5 is a right triangle. No, because a Pythagorean triple is a group of three , 7, and 14 is not a right triangle. No, because a Pythagorean triple is a group of three 2 2 2 10-5whole The Pythagorean Theorem numbers that satisfy the equation c = a + b , where c is the greatest number. 24. 8, 31.5, 32.5 26. 18, 24, 30 SOLUTION: Since the measure of the longest side is 32.5, let c = 2 32.5, a = 8, and b = 31.5. Then determine whether c 2 , , and is not a right triangle. No, because a Pythagorean triple is a group of three 2 2 2 whole numbers that satisfy the equation c = a + b , where c is the greatest number. 2 =a +b . 2 2 2 Yes, because c = a + b , a triangle with side lengths 8, 31.5, and 32.5 is a right triangle. No, because a Pythagorean triple is a group of three 2 2 2 whole numbers that satisfy the equation c = a + b , where c is the greatest number, but b and c are not whole numbers. SOLUTION: Since the measure of the longest side is 30, let c = 2 30, a = 18, and b = 24. Then determine whether c = 2 2 a +b . 2 2 2 Yes, because c = a + b , a triangle with side lengths 18, 24, and 30 is a right triangle. Yes, because a Pythagorean triple is a group of three 2 2 2 whole numbers that satisfy the equation c = a + b , where c is the greatest number. 27. 36, 77, 85 25. SOLUTION: Since the measure of the longest side is 85, let c = SOLUTION: 2 Since the measure of the longest side is ,a = , and b = 2 2 , let c = . Then determine 85, a = 36, and b = 77. Then determine whether c = 2 2 a +b . 2 whether c = a + b . 2 2 2 2 No, because c ≠ a + b , a triangle with side lengths , , and is not a right triangle. No, because a Pythagorean triple is a group of three 2 2 2 whole numbers that satisfy the equation c = a + b , where c is the greatest number. 26. 18, 24, 30 SOLUTION: Since the measure of the longest side is 30, let c = 2 30, a = 18, and b = 24. Then determine whether c = 2 2 a +b . 2 2 Yes, because c = a + b , a triangle with side lengths 36, 77, and 85 is a right triangle. Yes, because a Pythagorean triple is a group of three 2 2 2 whole numbers that satisfy the equation c = a + b , where c is the greatest number. 28. 17, 33, 98 SOLUTION: Since the measure of the longest side is 98, let c = 98, a = 17, and b = 33. The length of c is greater than the sum of a and b, so this is not a triangle. 2 2 2 No, because c ≠ a + b , a triangle with side lengths 17, 33, and 98 is not a triangle. No, because a Pythagorean triple is a group of three 2 2 2 whole numbers that satisfy the equation c = a + b , where c is the greatest number. 29. GEOMETRY Refer to the triangle shown. eSolutions Manual - Powered by Cognero 2 2 2 Yes, because c = a + b , a triangle with side lengths 18, 24, and 30 is a right triangle. Page 6 No, because c ≠ a + b , a triangle with side lengths 17, 33, and 98 is not a triangle. No, because a Pythagorean triple is a group of three 2 2 2 10-5whole The Pythagorean Theorem numbers that satisfy the equation c = a + b , where c is the greatest number. 29. GEOMETRY Refer to the triangle shown. a. What is a? b. Find the area of the triangle. 2 2 2 8 + 15 = 17 so the pieces form a right triangle by the converse of the Pythagorean Theorem. 31. LADDER Mr. Takeo is locked out of his house. The only open window is on the second floor. There is a bush along the edge of the house, so he places the neighbor’s ladder 10 feet from the house. To the nearest foot, what length of ladder does he need to reach the window? SOLUTION: a. Use the Pythagorean Theorem, substituting 11 for b and 23 for c. SOLUTION: Use the Pythagorean Theorem, substituting 10 for a and 28 for b. b. 2 He will need a 30-ft ladder to reach the window. The area of the triangle is about 111.1 units . CCSS TOOLS Find the length of the hypotenuse. Round to the nearest hundredth. 30. GARDENING Khaliah wants to plant flowers in a triangular plot. She has three lengths of plastic garden edging that measure 8 feet, 15 feet, and 17 feet. Determine whether these pieces form a right triangle. Explain. SOLUTION: Yes; sample answer: The longest side is 17 feet so this has to be the hypotenuse. 32. SOLUTION: Use the Pythagorean Theorem, substituting 7 for a and 10 for b. 2 2 2 8 + 15 = 17 so the pieces form a right triangle by the converse of the Pythagorean Theorem. 31. LADDER Mr. Takeo is locked out of his house. The only open window is on the second floor. There is a bush along the edge of the house, so he places the neighbor’s ladder 10 feet from the house. To the nearest foot, what length of ladder does he need to reach the window? eSolutions Manual - Powered by Cognero Page 7 10-5 The Pythagorean Theorem The height of the roof is 10.6 in. 35. GEOMETRY Each side of a cube is 5 inches long. Find the length of a diagonal of the cube. SOLUTION: 33. SOLUTION: Use the Pythagorean Theorem, substituting 4 for a and 7 for b. 34. DOLLHOUSE Alonso is building a dollhouse for his sister’s birthday. The roof is 24 inches across and the slanted side is 16 inches long as shown. Find the height of the roof to the nearest tenth of an inch. Use the Pythagorean Theorem to find the diagonal of a side of the cube, substituting 5 for a and 5 for b. Then, use the Pythagorean Theorem to find a diagonal of the cube, substituting 5 for a and for b. SOLUTION: Use the Pythagorean Theorem, substituting 12 for b and 16 for c. The length of a diagonal of the cube is about 8.66 in. The height of the roof is 10.6 in. 35. GEOMETRY Each side of a cube is 5 inches long. Find the length of a diagonal of the cube. SOLUTION: 36. TOWN SQUARES The largest town square in the world is Tiananmen Square in Beijing, China, covering 98 acres. a. One square mile is 640 acres. Assuming that Tiananmen Square is a square, how many feet long is a side to the nearest foot? b. To the nearest foot, what is the diagonal distance across Tiananmen Square? SOLUTION: a. First, use the conversion factor 1 square mile = 640 acres to convert the area of the square from acres to square miles. eSolutions Manual - Powered by Cognero Use the Pythagorean Theorem to find the diagonal of a side of the cube, substituting 5 for a and 5 for b. Page 8 So, 98 acres is 0.153125 square miles. Use 10-5 The Pythagorean Theorem The length of a diagonal of the cube is about 8.66 in. 36. TOWN SQUARES The largest town square in the world is Tiananmen Square in Beijing, China, covering 98 acres. a. One square mile is 640 acres. Assuming that Tiananmen Square is a square, how many feet long is a side to the nearest foot? b. To the nearest foot, what is the diagonal distance across Tiananmen Square? The diagonal distance across Tiananmen Square is about 2922 feet. 37. TRUCKS Violeta needs to construct a ramp to roll a cart of moving boxes from her garage into the back of her truck. The truck is 6 feet from the garage. The back of the truck is 36 inches above the ground. How long does the ramp have to be? SOLUTION: First, rewrite 36 inches as 3 ft. Use the Pythagorean thereom, substituting 6 for a and 3 for b. SOLUTION: a. First, use the conversion factor 1 square mile = 640 acres to convert the area of the square from acres to square miles. The ramp must be about 6.7 ft. So, 98 acres is 0.153125 square miles. Use dimensional analysis to convert this measure to square feet. If c is the measure of the hypotenuse of a right triangle, find each missing measure. If necessary, round to the nearest hundredth. 38. a = x, b = x + 41, c = 85 SOLUTION: Next, use the formula for the area of a square to find the length of a side. Use the Zero Product Property to solve for x. a = 36; b = 77 Therefore, a side of Tiananmen Square is about 2066 feet long. b. Use the Pythagorean Theorem to find the length of the diagonal of the square, substituting 2066 for both a and b. 39. a = 8, b = x, c = x + 2 SOLUTION: b = 15; c = 17 40. a = 12, b = x − 2, c = x The diagonal distance across Tiananmen Square is about 2922 feet. 37. TRUCKS Violeta needs to construct a ramp to roll a cart of moving boxes from her garage into the back of her truck. The truck is 6 feet from the garage. eSolutions Manual - Powered by Cognero The back of the truck is 36 inches above the ground. How long does the ramp have to be? SOLUTION: Page 9 b = 35; c = 37 10-5 The Pythagorean Theorem b = 15; c = 17 40. a = 12, b = x − 2, c = x SOLUTION: Use the Zero Product Property to solve for x. a = 9; b = 40; c = 41 44. GEOMETRY A right triangle has one leg that is 8 inches shorter than the other leg. The hypotenuse is 30 inches long. Find the length of each leg. SOLUTION: b = 35; c = 37 41. a = x, b = x + 7, c = 97 SOLUTION: Use the Zero Product Property to solve for x. a = 65; b = 72 42. a = x − 47, b = x, c = x + 2 SOLUTION: Use the Zero Product Property to solve for x. a = 16; b = 63; c = 65 43. a = x − 32, b = x − 1, c = x SOLUTION: Use the Zero Product Property to solve for x. a = 9; b = 40; c = 41 44. GEOMETRY A right triangle has one leg that is 8 inches shorter than the other leg. The hypotenuse is 30 inches long. Find the length of each leg. SOLUTION: eSolutions Manual - Powered by Cognero The length of each leg of the triangle is about 24.83 in. and about 16.83 in. 45. MULTIPLE REPRESENTATIONS In this problem, you will derive a method for finding the midpoint and length of a segment on the coordinate plane. a. Graphical Use a graph to find the lengths of the segments between (3, 2) and (8, 2) and between (4, 1) and (4, 9). Then find the midpoint of each segment. b. Logical Use what you learned in part a to write expressions for the lengths of the segments between (x1 , y) and (x2 , y) and between (x, y 1 ) and (x, y 2). What would be the midpoint of each segment? c. Analytical Based on your results from part b, find the midpoint of the segment with endpoints at (x1, y 1), and (x2, y 2). d. Analytical Use the Pythagorean Theorem to write an expression for the distance between (x1, y 1), and (x2, y 2). SOLUTION: a. From the graph, the length of the vertical segment is 9 – 1 or 8 units and the length of the horizontal line is 8 – 3 or 5 units. Page 10 and (x2, y 2). SOLUTION: From the graph, the length of the vertical segment 10-5a. The Pythagorean Theorem is 9 – 1 or 8 units and the length of the horizontal line is 8 – 3 or 5 units. 2 2 2 We know the Pythagorean Theorem is a + b = c . We can solve this formula for c by taking the square root of each side. So, . Now we can replace the variables with the lengths of each side. The length of side a is x2 – x1. The length of side b is y 2 – y 1. The length of side c is now . The midpoints will be located in the middle of each segment, or the average of the coordinates. For the vertical segment, the value between 1 and 9 is 5, so the midpoint should be at (4, 5). For the horizontal segment, the value between 3 and 8 is 5.5, so the midpoint should be at (5.5, 2). b. Distance: The length of the segment between (x1, y) and (x2, y) will be the absolute value of x1 – x2. 46. ERROR ANALYSIS Wyatt and Dario are determining whether 36, 77, and 85 form a Pythagorean triple. Is either of them correct? Explain your reasoning. The length of the segment between (x, y 1) and (x, y2) will be the absolute value of y 1 – y 2. Midpoint: When the x-coordinates are identical, then the x-coordinate of the midpoint will also be the same value. This is true for the y-coordinates as well. The midpoint of (x1, y) and (x2, y) is at the average of the x-coordinates and y or at , and the midpoint of (x, y 1) and (x, y 2) is at . c. When the x- and y-coordinates are both different, the midpoint should be at the average values of the . coordinates, or at d. Form a right triangle with the segment formed by the two points as the hypotenuse. Note that the other coordinate is (x2, y 1) in the drawing. 2 2 2 We know the Pythagorean Theorem is a + b = c . eSolutions Manual - Powered by Cognero We can solve this formula for c by taking the square root of each side. So, . Now we can replace the variables with the lengths of each SOLUTION: Wyatt is correct. The square of the greatest value should be equal to the sum of the squares of the two smaller values. Since this is the case, the numbers form a Pythagorean triple. 47. CCSS PERSEVERANCE Find the value of x in the figure shown. SOLUTION: The figure can be separated into two right triangles. Let m represent the missing side length common to both right triangles. From the Pythagorean Theorem, 2 2 2 2 2 2 m = 2 + x and 14 = 8 + m . Using substitution, 2 2 2 2 you can find that 2 + x = 14 – 8 . Solve for x. Page 11 SOLUTION: Wyatt is correct. The square of the greatest value should be equal to the sum of the squares of the two 10-5smaller values. Since this is the case, the numbers The Pythagorean Theorem form a Pythagorean triple. 47. CCSS PERSEVERANCE Find the value of x in the figure shown. area of 6 cm . A right triangle with legs measuring 2 cm and cm also has a hypotenuse of 5 cm, but its area is 2 cm , which is not equivalent to 6 2 cm . 49. OPEN ENDED Draw a right triangle that has a hypotenuse of units. SOLUTION: Use the Pythagorean Theorem to select two side lengths for a and b with the given value of c. SOLUTION: The figure can be separated into two right triangles. Let m represent the missing side length common to both right triangles. From the Pythagorean Theorem, 2 2 2 2 2 2 m = 2 + x and 14 = 8 + m . Using substitution, 2 2 2 2 you can find that 2 + x = 14 – 8 . Solve for x. 50. WRITING IN MATH Explain how to determine whether segments in three lengths could form a right triangle. 48. REASONING Provide a counterexample to the statement. Any two right triangles with the same hypotenuse have the same area. SOLUTION: From the converse of the Pythagorean Theorem, if 2 2 2 a + b = c then a, b, and c are the lengths of the side of a right triangle. So, check to see whether the square of the greatest number is equal to the sum of the squares of the other two numbers. Consider triangle with sides of 9, 12 and 15 and a triangle with sides of 7, 13 and 19. SOLUTION: Different lengths of a and b can produce the same value of c in the Pythagorean theorem. 51. GEOMETRY Find the missing length. Sample answer: A right triangle with legs measuring 3 cm and 4 cm has a hypotenuse of 5 cm and an 2 area of 6 cm . A right triangle with legs measuring 2 cm and cm also has a hypotenuse of 5 cm, but its area is 2 cm , which is not equivalent to 6 2 cm . A −17 B − C D 17 SOLUTION: 49. OPEN ENDED Draw a right triangle that has a hypotenuse of units. eSolutions Manual - Powered by Cognero SOLUTION: Use the Pythagorean Theorem to select two side Page 12 10-5 The Pythagorean Theorem The correct choice is H. 51. GEOMETRY Find the missing length. 53. SHORT RESPONSE A plumber charges $40 for the first hour of each house call plus $8 for each additional half hour. If the plumber works for 4 hours, how much does he charge? SOLUTION: Let x = the total number of hours worked. A −17 B − C D 17 SOLUTION: The plumber charges $88 for 4 hours of work. 54. Find the next term in the geometric sequence . A B The correct choice is C. 52. What is a solution of this equation? C D F 0, 3 G 3 H 0 J no solutions SOLUTION: SOLUTION: The correct choice is B. The correct choice is H. Solve each equation. Check your solution. 53. SHORT RESPONSE A plumber charges $40 for the first hour of each house call plus $8 for each additional half hour. If the plumber works for 4 hours, how much does he charge? 55. SOLUTION: SOLUTION: Let x = the total number of hours worked. Check. eSolutions - Powered by $88 Cognero The Manual plumber charges for 4 hours of work. 54. Find the next term in the geometric Page 13 56. 10-5 The Pythagorean Theorem The correct choice is B. Solve each equation. Check your solution. 58. 55. SOLUTION: SOLUTION: Check. Check. 56. SOLUTION: 59. SOLUTION: Check. Check. 57. SOLUTION: 60. SOLUTION: Check. Check. 58. SOLUTION: There is no real solution. Simplify each expression. eSolutions Manual - Powered by Cognero 61. Check. SOLUTION: Page 14 10-5 The Pythagorean Theorem There is no real solution. Simplify each expression. 61. Describe how the graph of each function is 2 related to the graph of f (x) = x . 2 SOLUTION: 67. g(x) = x − 8 SOLUTION: 2 The graph of f (x) = x + c represents a vertical translation of the parent graph. The value of c is –8, 2 and –8 < 0. If c < 0, the graph of f (x) = x is translated units down. Therefore, the graph of g 62. SOLUTION: 2 (x) = x – 8 is a translation of the parent graph shifted down 8 units. 68. h(x) = 63. SOLUTION: x 2 SOLUTION: 2 The graph of f (x) = ax stretches or compresses the 2 graph of f (x) = x vertically. The value of a is < 1. If 0 < and 0 < , < 1, the graph of f (x) = x 2 is compressed vertically. Therefore, the graph of h 64. (x) = SOLUTION: 2 x is the parent graph compressed vertically. 2 69. h(x) = −x + 5 SOLUTION: 2 The graph of f (x) = –x reflects the graph of f (x) = 2 2 x across the x-axis. The graph of f (x) = x + c represents a vertical translation of the parent graph. The value of c is 5, and 5 > 0. If c > 0, the graph of f 2 (x) = x is translated units up. Therefore, the 65. SOLUTION: 2 graph of h(x) = −x + 5 is a translation of the parent graph shifted up 8 units and reflected across the xaxis. 2 70. g(x) = (x + 10) 66. SOLUTION: SOLUTION: 2 The graph of f (x) = (x – c) represents a horizontal translation of the parent graph. The value of c is –10, 2 and –10 < 0. If c < 0, the graph of f (x) = (x – c) is translated units left. Therefore, the graph of g(x) 2 Describe how the graph of each function is 2 related to the graph of f (x) = x . = (x + 10) is a translation of the parent graph shifted left 10 units. eSolutions Manual - Powered by Cognero 2 67. g(x) = x − 8 71. g(x) = −2x 2 SOLUTION: Page 15 2 Therefore, the graph of h(x) = −x units up. Therefore, the (x) = x is translated 2 is a 2 graph of h(x) = −x + 5 is a translation of the parent shifted up 8 units and reflected across the x10-5graph The Pythagorean Theorem axis. 2 70. g(x) = (x + 10) SOLUTION: 2 The graph of f (x) = (x – c) represents a horizontal translation of the parent graph. The value of c is –10, 2 and –10 < 0. If c < 0, the graph of f (x) = (x – c) is translated units left. Therefore, the graph of g(x) units and reflected across the x-axis. 73. ROCK CLIMBING While rock climbing, Damaris launches a grappling hook from a height of 6 feet with an initial upward velocity of 56 feet per second. The hook just misses the stone ledge that she wants to scale. As it falls, the hook anchors on a ledge 30 feet above the ground. How long was the hook in the air? SOLUTION: 2 = (x + 10) is a translation of the parent graph shifted left 10 units. 71. g(x) = −2x translation of the parent graph shifted down 2 SOLUTION: 2 The graph of f (x) = –x reflects the graph of f (x) = 2 2 x across the x-axis. The graph of f (x) = ax 2 stretches or compresses the graph of f (x) = x vertically. The value of a is –2, and |–2| > 1. If > 2 1, the graph of f (x) = x is stretched vertically. 2 Therefore, the graph of g(x) = –2x is the parent graph reflected across the x-axis and stretched vertically. A half second after she throws the hook, it is 30 feet in the air. The hook continues up, then drops back down to the height of 30 feet 3 seconds after the throw. The hook was in the air for 3 seconds. Find each product. 74. (b + 8)(b + 2) SOLUTION: 2 72. h(x) = −x − SOLUTION: 2 The graph of f (x) = –x reflects the graph of f (x) = 2 2 x across the x-axis. The graph of f (x) = x + c represents a vertical translation of the parent graph. The value of c is , and SOLUTION: < 0. If c < 0, the 2 graph of f (x) = x is translated units down. Therefore, the graph of h(x) = −x 2 76. (y + 4)(y − 8) SOLUTION: is a translation of the parent graph shifted down units and reflected across the x-axis. 73. ROCK CLIMBING While rock climbing, Damaris launches a grappling hook from a height of 6 feet with an initial upward velocity of 56 feet per second. The hook just misses the stone ledge that she wants to scale. As it falls, the hook anchors on a ledge 30 feet above the ground. How long was the hook in the air? SOLUTION: 75. (x − 4)(x − 9) 77. (p + 2)(p − 10) SOLUTION: 78. (2w − 5)(w + 7) SOLUTION: eSolutions Manual - Powered by Cognero Page 16 79. (8d + 3)(5d + 2) 77. (p + 2)(p − 10) SOLUTION: 10-5 The Pythagorean Theorem 78. (2w − 5)(w + 7) 82. SOLUTION: SOLUTION: 79. (8d + 3)(5d + 2) SOLUTION: 83. 80. BUSINESS The amount of money spent at West Outlet Mall continues to increase. The total T(x) in millions of dollars can be estimated by the function T SOLUTION: x (x) = 12(1.12) , where x is the number of years after it opened in 2005. Find the amount of sales in 2015, 2016, and 2017. SOLUTION: 84. SOLUTION: The sales for the mall in 2015 will be about $37.27 million; in 2016 will be about $41.74 million; and in 2017 will be about $46.75 million. Solve each proportion. 81. SOLUTION: 82. SOLUTION: eSolutions Manual - Powered by Cognero 83. Page 17 10-6 Trigonometric Ratios Find the values of the three trigonometric ratios for angle A . 3. 1. SOLUTION: SOLUTION: 4. SOLUTION: 2. SOLUTION: CCSS TOOLS Use a calculator to find the value of each trigonometric ratio to the nearest tenthousandth. 5. sin 37° SOLUTION: 3. SOLUTION: eSolutions Manual - Powered by Cognero Keystrokes: 37 Then, sin 37° = 0.6018. 6. cos 23° SOLUTION: Page 1 SOLUTION: : Ratios 37 10-6Keystrokes Trigonometric Then, sin 37° = 0.6018. 6. cos 23° SOLUTION: 10. Keystrokes: 23 Then, cos 23° = 0.9205 SOLUTION: Find the measure of 7. tan 14° . SOLUTION: Keystrokes: 14 Then, tan 14° = 0.2493. Find . 8. cos 82° SOLUTION: Keystrokes: 82 Then, cos 82° = 0.1392. Find . Solve each right triangle. Round each side length to the nearest tenth. 9. SOLUTION: Find the measure of . SOLUTION: Find the measure of Find 11. . . Find . Find . Find . 10. SOLUTION: Find the measure of eSolutions Manual - Powered by Cognero . Page 2 10-6 Trigonometric Ratios The length of the run of the hill is about 11,326.2 ft. Find m X for each right triangle to the nearest degree. 12. SOLUTION: Find the measure of 14. . SOLUTION: –1 Use tan Find on a calculator. Keystrokes: . -1 [TAN ] Find . 15. SOLUTION: 13. SNOWBOARDING A hill used for snowboarding has a vertical drop of 3500 feet. The angle the run makes with the ground is 18°. Estimate the length of r. –1 Use cos on a calculator. Keystrokes: -1 [COS ] SOLUTION: The length of the run of the hill is about 11,326.2 ft. Find m X for each right triangle to the nearest degree. 16. SOLUTION: –1 Use tan on a calculator. Keystrokes: -1 [TAN ] 14. SOLUTION: –1 Use Manual tan on a calculator. eSolutions - Powered by Cognero Keystrokes: -1 [TAN ] Page 3 Keystrokes: Keystrokes: [TAN ] [SIN ] 10-6 Trigonometric Ratios Find the values of the three trigonometric ratios for angle B. 17. SOLUTION: Use sin –1 18. on a calculator. Keystrokes: SOLUTION: Find -1 [SIN ] . Find the values of the three trigonometric ratios for angle B. 18. SOLUTION: Find . 19. SOLUTION: Find . eSolutions Manual - Powered by Cognero Page 4 10-6 Trigonometric Ratios 19. 20. SOLUTION: Find . SOLUTION: Find . CCSS TOOLS Use a calculator to find the value of each trigonometric ratio to the nearest tenthousandth. 21. tan 2° 20. SOLUTION: SOLUTION: Find Keystrokes: 2 Then, tan 2° = 0.0349. . 22. sin 89° SOLUTION: Keystrokes: 89 Then, sin 89° = 0.9998. eSolutions Manual - Powered by Cognero Page 5 23. cos 44° SOLUTION: 28. tan 60° SOLUTION: SOLUTION: Keystrokes: 2 10-6Then, tan 2° = 0.0349. Trigonometric Ratios Keystrokes: 60 Then, tan 60° = 1.7321. 22. sin 89° Solve each right triangle. Round each side length to the nearest tenth. SOLUTION: Keystrokes: 89 Then, sin 89° = 0.9998. 23. cos 44° 29. SOLUTION: SOLUTION: Find the measure of Keystrokes: 44 Then, cos 44° = 0.7193. 24. tan 45° Find SOLUTION: Keystrokes: Then, tan 45° = 1. . . 45 25. sin 73° SOLUTION: Keystrokes: 73 Then, sin 73° = 0.9563 Find . 26. cos 90° SOLUTION: Keystrokes: Then, cos 90° = 0. 90 27. sin 30° 30. SOLUTION: SOLUTION: Find the measure of Keystrokes: 30 Then, sin 30° = 0.5. . 28. tan 60° SOLUTION: Find . Keystrokes: 60 Then, tan 60° = 1.7321. Solve each right triangle. Round each side length to the nearest tenth. Find . 29. SOLUTION: Find the measure of . eSolutions Manual - Powered by Cognero Page 6 31. 10-6 Trigonometric Ratios 31. SOLUTION: Find the measure of . 33. SOLUTION: Find the measure of Find . . Find . Find . Find . 32. SOLUTION: Find the measure of . 34. SOLUTION: Find the measure of . Find . Find . Find . Find eSolutions Manual - Powered by Cognero . Page 7 35. ESCALATORS At a local mall, an escalator is 110 feet long. The angle the escalator makes with the 10-6 Trigonometric Ratios The escalator is about 53 ft high. Find m J for each right triangle to the nearest degree. 34. SOLUTION: Find the measure of 36. . SOLUTION: You know the measure of the side opposite ∠J and the measure of the hypotenuse. Use the sine ratio. Find . –1 Use a calculator and the [sin ]function to find the measure of the angle. Keystrokes: Find . -1 [SIN ] 10 24 24.62431835 So, m∠J ≈ 25°. 35. ESCALATORS At a local mall, an escalator is 110 feet long. The angle the escalator makes with the ground is 29°. Find the height reached by the escalator. 37. SOLUTION: SOLUTION: You know the measure of the side opposite ∠J and the measure of the hypotenuse. Use the sine ratio. –1 Use a calculator and the [sin ] function to find the measure of the angle. The escalator is about 53 ft high. Find m J for each right triangle to the nearest degree. Keystrokes: -1 [SIN ] 15 17 61.92751306 So, m∠J ≈ 62°. 36. SOLUTION: eSolutions Powered by Cognero YouManual know -the measure of the side opposite ∠J and the measure of the hypotenuse. Use the sine ratio. Page 8 61.92751306 So, m∠J ≈ 62°. 10-6 Trigonometric Ratios 30.96375653 So, m∠J ≈ 31°. 40. SOLUTION: You know the measure of the side adjacent to ∠J and the measure of the hypotenuse. Use the cosine ratio. 38. SOLUTION: You know the measure of the side opposite ∠J and the measure of the side adjacent to ∠J. Use the tangent ratio. –1 Use a calculator and the [cos ] function to find the measure of the angle. –1 Use a calculator and the [tan ] function to find the measure of the angle. Keystrokes: Keystrokes: -1 [TAN ] 23 -1 [COS ] 5 16 71.79004314 14 So, m∠J ≈ 72°. 58.67130713 So, m∠J ≈ 59°. 41. 39. SOLUTION: You know the measure of the side opposite ∠J and the measure of the side adjacent to ∠J. Use the tangent ratio. SOLUTION: You know the measure of the side adjacent to ∠J and the measure of the hypotenuse. Use the cosine ratio. –1 Use a calculator and the [tan ] function to find the measure of the angle. –1 Use a calculator and the [cos ] function to find the measure of the angle. Keystrokes: -1 [TAN ] 6 10 30.96375653 Keystrokes: -1 [COS ] 11 So, m∠J ≈ 31°. 49.67978493 So, m∠J ≈ 50°. 40. SOLUTION: eSolutions Manual - Powered by Cognero You know the measure of the side adjacent to ∠J and the measure of the hypotenuse. Use the cosine 17 42. MONUMENTS The Lincoln Memorial building measures 204 feet long, 134 feet wide, and 99 feet tall. Chloe is looking at the top of the monument at an angle of 55°. How far away is she standing from the monument? Page 9 SOLUTION: 49.67978493 m∠J ≈ 50°. Ratios 10-6So, Trigonometric 42. MONUMENTS The Lincoln Memorial building measures 204 feet long, 134 feet wide, and 99 feet tall. Chloe is looking at the top of the monument at an angle of 55°. How far away is she standing from the monument? SOLUTION: The horizontal distance to the city is about 35,577 ft. 44. FORESTS A forest ranger estimates the height of a tree is about 175 feet. If the forest ranger is standing 100 feet from the base of the tree, what is the measure of the angle formed between the range and the top of the tree? SOLUTION: The angle formed between the ground and the top of the tree is about 60°. Chloe is standing about 69 ft away from the monument. 43. AIRPLANES Ella looks down at a city from an airplane window. The airplane is 5000 feet in the air, and she looks down at an angle of 8°. Determine the horizontal distance to the city. Suppose ABC . A is an acute angle of right triangle 45. Find sin A and tan A if cos . SOLUTION: SOLUTION: The horizontal distance to the city is about 35,577 ft. 44. FORESTS A forest ranger estimates the height of a tree is about 175 feet. If the forest ranger is standing 100 feet from the base of the tree, what is the measure of the angle formed between the range and the top of the tree? eSolutions Manual - Powered by Cognero SOLUTION: 46. Find tan A and cos A if sin SOLUTION: . Page 10 10-6 Trigonometric Ratios 46. Find tan A and cos A if sin . . SOLUTION: SOLUTION: 47. Find cos A and tan A if sin 48. Find sin A and cos A if tan . SOLUTION: 49. SUBMARINES A submarine descends into the ocean at an angle of 10° below the water line and travels 3 miles diagonally. How far beneath the surface of the water has the submarine reached? SOLUTION: The submarine went about 0.5 mi beneath the surface of the water. 48. Find sin A and cos A if tan . 50. MULTIPLE REPRESENTATIONS In this problem, you will explore a relationship between the sine and cosine functions. SOLUTION: a. TABULAR Copy and complete the table using the triangles shown above. eSolutions Manual - Powered by Cognero Page 11 b. VERBAL Make a conjecture about the sum of 10-6 Trigonometric Ratios a. TABULAR Copy and complete the table using the triangles shown above. b. The sum of the squares of the sine and cosine of an acute angle in a right triangle is equal to 1. 51. CHALLENGE Find a and c in the triangle shown. b. VERBAL Make a conjecture about the sum of the squares of the sine and cosine of an acute angle of an acute angle in a right triangle. SOLUTION: a. SOLUTION: Use the sum of the angles of a triangle to determine the value of a. Use the value of a to determine the value of the angles of the triangle. The angles are 90°, 6(5) – 3 or 27°, and 12(5) + 3 or 63°. Use a trigonometric ratio to find the value of c. Therefore, a = 5 and c ≈ 7.3. Then , 52. REASONING Use the definitions of the sine and cosine ratios to define the tangent ratio. SOLUTION: Sin is defined as and Cos is defined as Tan can be defined as . because: b. The sum of the squares of the sine and cosine of an acute angle in a right triangle is equal to 1. 51. CHALLENGE Find a and c in the triangle shown. eSolutions Manual - Powered by Cognero Page 12 Therefore, a = 5 and c ≈ 7.3. 10-6 Trigonometric Ratios 52. REASONING Use the definitions of the sine and cosine ratios to define the tangent ratio. SOLUTION: Sin is defined as and Cos is defined as Tan can be defined as . 54. CCSS ARGUMENTS The sine and cosine of an acute angle in a right triangle are equal. What can you conclude about the triangle? SOLUTION: Given: ΔABC with sides a, b, and c as shown; sin A = cos A because: 53. WRITING IN MATH How can triangles be used to solve problems? SOLUTION: Many real world problems involve trying to determine the correct height or length of a given structure. When lengths and angles are known, right triangles can be drawn, and trigonometric ratios can be used to determine missing sides and angles. Similarly, other situations may require triangles and the Pythagorean theorem to determine unknown lengths. If a = b, then . A triangle that has two congruent sides is called an isosceles triangle. Therefore, this triangle is an isosceles right triangle. The legs of the right triangle are equal to each other. 55. WRITING IN MATH Explain how to use trigonometric ratios to find the missing length of a side of a right triangle given the measure of one acute angle and the length of one side. SOLUTION: Use the angle given and the measure of the known side to set up one of the trigonometric ratios. The sine ratio uses the opposite side and hypotenuse of the triangle. The cosine ratio uses the adjacent side and hypotenuse of the triangle. The tangent ratio uses the opposite and adjacent sides of the triangle. Choose the ratio that can be used to solve for the unknown measure. Given the following triangle find the missing sides a and c. 54. CCSS ARGUMENTS The sine and cosine of an acute angle in a right triangle are equal. What can you conclude about the triangle? eSolutions Manual - Powered by Cognero SOLUTION: Given: ΔABC with sides a, b, and c as shown; sin A = cos A Page 13 Since you know the measure of ∠A, set up the uses the opposite and adjacent sides of the triangle. Choose the ratio that can be used to solve for the unknown measure. 10-6 Trigonometric Ratios Given the following triangle find the missing sides a and c. 56. Which graph below represents the solution set for −2 ≤ x ≤ 4? A B C Since you know the measure of ∠A, set up the trigonometric ratios for the acute angle of 42°. Let a be the measure of the side opposite ∠A, 15 is the measure of the side adjacent ∠A, and c is the measure of the hypotenuse. So, if you are trying to find the measure of a, use the tangent ratio. If you are trying to find the measure of c, use the cosine ratio. D SOLUTION: The inequality uses less than or equal to signs, so the points on the graph must be solid. So, choices B and D are incorrect. In the inequality, x is found between the two values, so choice C in incorrect. The correct choice is A. 57. PROBABILITY Suppose one chip is chosen from a bin with the chips shown. To the nearest tenth, what is the probability that a green chip is chosen? F 0.2 G 0.5 H 0.6 J 0.8 56. Which graph below represents the solution set for −2 ≤ x ≤ 4? A SOLUTION: B C The correct choice is F. 58. In the graph, for what value(s) of x is y = 0? D SOLUTION: eSolutions Manual - Powered by Cognero The inequality uses less than or equal to signs, so the points on the graph must be solid. So, choices B and Page 14 10-6 Trigonometric Ratios The correct choice is F. a. Let h represent the height reached by the ladder. Use the Pythagorean Theorem to represent the value of h in terms of the other two sides. 58. In the graph, for what value(s) of x is y = 0? If the bottom of the ladder is moved closer to the base of the house, the distance the bottom of the 2 A 0 B −1 C 1 D 1 and −1 SOLUTION: The graph crosses the x-axis twice, so there are two values of x for which y = 0. Choices A, B, and C only offer one x value. The correct choice is D. 59. EXTENDED RESPONSE A 16-foot ladder is placed against the side of a house so that the bottom of the ladder is 8 feet from the base of the house. a. If the bottom of the ladder is moved closer to the base of the house, does the height reached by the ladder increase or decrease? b. What conclusion can you make about the distance between the bottom of the ladder and the base of the house and the height reached by the ladder? c. How high does the ladder reach if the ladder is 3 feet from the base of the house? ladder is from the wall will decrease. When 16 is 2 subtracted by a number smaller than 8 , the 2 2 difference is greater than 16 - 8 . Since you are finding the square root of a larger number, h will be greater. Therefore, as the bottom of the ladder is moved closer to the base of the house, the height reached by the ladder will increase. b. Sample answer: Let h represent the height reached by the ladder and d represent the distance between the bottom of the ladder and the base of the house. The house is built perpendicular to the ground, so the ladder will form a right triangle when it is placed against the side of the house. Use the Pythagorean Theorem to relate the sides of the triangle. 2 Therefore, the sum of their squares is 16 or 256. c. SOLUTION: If the ladder is 3 ft from the base of the house, then it reaches a height of about 15.7 ft. If c is the measure of the hypotenuse of a right triangle, find each missing measure. If necessary, round to the nearest hundredth. 60. a = 16, b = 63, c = ? SOLUTION: a. Let h represent the height reached by the ladder. Use the Pythagorean Theorem to represent the value of h in terms of the other two sides. eSolutions Manual - Powered by Cognero Page 15 The length of the hypotenuse is 65 units. If the bottom of the ladder is moved closer to the The length of one of the legs is units. the ladder is 3 ftRatios from the base of the house, then it 10-6IfTrigonometric reaches a height of about 15.7 ft. If c is the measure of the hypotenuse of a right triangle, find each missing measure. If necessary, round to the nearest hundredth. 60. a = 16, b = 63, c = ? or about 10.72 63. a = 6, b = 3, c = ? SOLUTION: SOLUTION: The length of the hypotenuse is units. The length of the hypotenuse is 65 units. 64. 61. b = 3, ,c=? or about 6.71 , c = 12, a = ? SOLUTION: SOLUTION: The length of the hypotenuse is 11 units. The length of one of the legs is units. or about 8.19 62. c = 14, a = 9, b = ? 65. a = 4, SOLUTION: The length of one of the legs is units. 63. a = 6, b = 3, c = ? SOLUTION: ,c=? SOLUTION: or about 10.72 The length of the hypotenuse is units. or about 5.20 66. AVIATION The relationship between a plane’s length L in feet and the pounds P its wings can lift is eSolutions Manual - Powered by Cognero described by , where k is the constant of proportionality. A Boeing 747 is 232 feet long and has a takeoff weight of 870,000 pounds. Determine k Page 16 for this plane to the nearest hundredth. SOLUTION: length of the hypotenuse is 10-6The Trigonometric Ratios units. or about 5.20 66. AVIATION The relationship between a plane’s length L in feet and the pounds P its wings can lift is 69. described by , where k is the constant of proportionality. A Boeing 747 is 232 feet long and has a takeoff weight of 870,000 pounds. Determine k for this plane to the nearest hundredth. SOLUTION: SOLUTION: 70. The constant of proportionality is about 0.06. SOLUTION: 67. FINANCIAL LITERACY A salesperson is paid $32,000 a year plus 5% of the amount in sales made. What is the amount of sales needed to have an annual income greater than $45,000? SOLUTION: Let x represent the amount of sales made. 71. The amount of sales must be more than $260,000. Solve each proportion. SOLUTION: 68. SOLUTION: 69. SOLUTION: eSolutions Manual - Powered by Cognero Page 17 . Another way to identify the stretch is to notice that the y-values in the table are 2 times the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}. Mid-Chapter Quiz Graph each function. Compare to the parent graph. State the domain and range. 2. SOLUTION: Make a table. x 0 0.5 1 2 3 y 0 ≈ – –4 ≈ – ≈ – 2.8 5.7 6.9 Plot the points and draw a smooth curve. 1. SOLUTION: Make a table. x 0 0.5 1 2 3 y 0 ≈ 1.4 2 ≈ 2.8 ≈ 3.5 Plot the points and draw a smooth curve. 4 4 4 –8 The parent function is multiplied by a value less than 1, so the graph is a vertical stretch of and a reflection across the x-axis. Another way to identify the stretch is to notice that the yvalues in the table are –4 times the corresponding yvalues for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}. is multiplied by a value The parent function greater than 1, so the graph is a vertical stretch of . Another way to identify the stretch is to notice that the y-values in the table are 2 times the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}. 3. 2. SOLUTION: Make a table. x 0 0.5 1 2 3 y 0 ≈ – –4 ≈ – ≈ – 2.8 5.7 6.9 Plot the points and draw a smooth curve. 4 –8 The parent function is multiplied by a value less than 1, so the graph is a vertical stretch of Another and a reflection across the x-axis. eSolutions Manual - Powered by Cognero way to identify the stretch is to notice that the yvalues in the table are –4 times the corresponding y- SOLUTION: Make a table. x 0 0.5 1 2 3 y 0 0.5 ≈ 0.4 ≈ 0.7 ≈ 0.9 Plot the points and draw a smooth curve. 4 1 The parent function is multiplied by a value less than 1 and greater than 0, so the the graph is a vertical compression of . Another way to Page 1 identify the compression is to notice that the y-values in the table are times the corresponding y-values and a reflection across the x-axis. Another way to identify the stretch is to notice that the yvalues in the table are –4 times the corresponding yvalues for Quiz the parent function. The domain is {x|x ≥ Mid-Chapter 0} and the range is {y|y ≤ 0}. identify the compression is to notice that the y-values in the table are times the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}. 4. 3. SOLUTION: Make a table. x 0 0.5 1 2 3 y 0 0.5 ≈ 0.4 ≈ 0.7 ≈ 0.9 Plot the points and draw a smooth curve. SOLUTION: Make a table. x 0 0.5 1 2 3 y –3 ≈ – –2 ≈ – ≈ – 2.3 1.6 1.3 Plot the points and draw a smooth curve. 4 1 4 –1 The parent function is multiplied by a value less than 1 and greater than 0, so the the graph is a vertical compression of . Another way to identify the compression is to notice that the y-values The value 3 is being subtracted from the parent function , so the graph is translated down 3 units from the parent graph . Another way to identify the translation is to note that the y-values in the table are 3 less than the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ –3}. in the table are times the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}. 5. 4. SOLUTION: Make a table. x 0 0.5 1 2 3 y –3 ≈ – –2 ≈ – ≈ – 2.3 1.6 1.3 Plot the points and draw a smooth curve. 4 –1 SOLUTION: Make a table x 1 1.5 2 3 y 0 1 ≈ 0.7 ≈ 1.4 Plot the points and draw a smooth curve. 4 ≈ 1.7 The value 3 is being subtracted from the parent function , so the graph is translated down 3 eSolutions Powered bygraph Cognero unitsManual from -the parent . Another way to identify the translation is to note that the y-values in the table are 3 less than the corresponding y-values The value 1 is being subtracted from the square root of the parent function , so the graph is translated 1 unit right from the parent graph . Another way to identify the translation is to notePage that 2 the x-values in the table are 1 more than the corresponding x-values for the parent function. The units from the parent graph . Another way to identify the translation is to note that the y-values in the table are 3 less than the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and Mid-Chapter Quiz the range is {y|y ≥ –3}. 5. translated 1 unit right from the parent graph . Another way to identify the translation is to note that the x-values in the table are 1 more than the corresponding x-values for the parent function. The domain is {x|x ≥ 1}, and the range is {y|y ≥ 0}. 6. SOLUTION: Make a table x 1 1.5 2 3 y 0 1 ≈ 0.7 ≈ 1.4 Plot the points and draw a smooth curve. SOLUTION: Make a table. x 2 2.5 3 4 y 0 2 ≈ 1.4 ≈ 2.8 Plot the points and draw a smooth curve. 4 ≈ 1.7 6 4 The value 1 is being subtracted from the square root of the parent function , so the graph is This graph is the result of a vertical stretch of the translated 1 unit right from the parent graph . Another way to identify the translation is to note that the x-values in the table are 1 more than the corresponding x-values for the parent function. The domain is {x|x ≥ 1}, and the range is {y|y ≥ 0}. graph of y = followed by a translation 2 units right. The domain is {x|x ≥ 2}, and the range is {y|y ≥ 0}. 7. GEOMETRY The length of the side of a square is given by the function , where A is the area of the square. What is the length of the side of a square that has an area of 121 square inches? A 121 inches B 44 inches C 11 inches D 10 inches 6. SOLUTION: Make a table. x 2 2.5 3 4 y 0 2 ≈ 1.4 ≈ 2.8 Plot the points and draw a smooth curve. 6 4 SOLUTION: The side length of the square is 11 inches. So C is the correct choice. Simplify each expression. 8. SOLUTION: This graph is the result of a vertical stretch of the graph of y = followed by a translation 2 units right. The domain is {x|x ≥ 2}, and the range is {y|y ≥ 0}. eSolutions Manual - Powered by Cognero 7. GEOMETRY The length of the side of a square is given by the function , where A is the area 9. Page 3 SOLUTION: Mid-Chapter Quiz 9. SOLUTION: 13. SATELLITES A satellite is launched into orbit 200 kilometers above Earth. The orbital velocity of a satellite is given by the formula . v is velocity in meters per second, G is a given constant, mE is the mass of Earth, and r is the radius of the satellite’s orbit in meters. a. The radius of Earth is 6,380,000 meters. What is the radius of the satellite’s orbit in meters? b. The mass of Earth is 5.97 × 1024 kilograms, and 10. SOLUTION: where N is in the constant G is Newtons. Use the formula to find the orbital velocity of the satellite in meters per second. SOLUTION: a. Convert 200 kilometers to meters. 11. SOLUTION: The satellite is orbiting 200,000 meters above Earth. The satellite’s orbit is equal to the sum of the radius of Earth and the distance that the satellite is from Earth. 6,380,000 m + 200,000 m = 6,580,000 m The radius of the satellite’s orbit is 6,580,000 meters. b. Substitute 6.77 × 10 –11 24 for G, 5.97 × 10 for mg and 6,580,000 for r. 12. SOLUTION: The velocity of the satellite is approximately 7779 meters per second. 14. Which expression is equivalent to eSolutions Manual - Powered by Cognero 13. SATELLITES A satellite is launched into orbit 200 kilometers above Earth. The orbital velocity of a ? Page 4 F 16. SOLUTION: The velocity of the satellite is approximately 7779 Mid-Chapter Quiz meters per second. 14. Which expression is equivalent to ? 17. SOLUTION: F G H 2 J 4 18. SOLUTION: SOLUTION: 19. SOLUTION: So, the correct choice is G. Simplify each expression. 20. 15. SOLUTION: SOLUTION: 16. SOLUTION: 21. SOLUTION: 17. SOLUTION: eSolutions Manual - Powered by Cognero 18. 22. GEOMETRY Find the area of the rectangle. Page 5 Mid-Chapter Quiz 22. GEOMETRY Find the area of the rectangle. 24. SOLUTION: SOLUTION: Substitute for , and for w. Check: 25. SOLUTION: The area of the rectangle is square units. Solve each equation. Check your solution. 23. SOLUTION: Check: Check: 26. SOLUTION: 24. SOLUTION: eSolutions Manual - Powered by Cognero Check: Page 6 Because –4 does not satisfy the original equation, 4 is the only solution. Mid-Chapter Quiz 27. 26. SOLUTION: SOLUTION: Check: Check: Because Because –4 does not satisfy the original equation, 4 is the only solution. does not satisfy the original equation, 5 is the only solution. 28. 27. SOLUTION: SOLUTION: Check: Check: 29. GEOMETRY The lateral surface area S of a cone eSolutions Manual - Powered by Cognero Because does not satisfy the original equation, 5 is the only solution. can be found by using the formula , Page 7 where r is the radius of the base and h is the height of the cone. Find the height of the cone. Mid-Chapter Quiz 29. GEOMETRY The lateral surface area S of a cone can be found by using the formula , where r is the radius of the base and h is the height of the cone. Find the height of the cone. SOLUTION: Substitute 121 for s and 3 for r. Then solve for h. The height of the cone is about 12.5 inches. eSolutions Manual - Powered by Cognero Page 8 across the x-axis. Another way to identify the graph i to notice that the y-values in the table are –1 times th corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}. Practice Test - Chapter 10 Graph each function, and compare to the parent graph. State the domain and range. 2. 1. SOLUTION: Make a table. x 0 0.5 1 2 3 y 0 ≈ – –1 ≈ – ≈ – 0.7 1.4 1.7 Plot the points and draw a smooth curve. SOLUTION: Make a table. x 0 0.5 1 2 3 y 0 ≈ 0.18 0.25 ≈ 0.35 ≈ 0.43 Plot the points and draw a smooth curve. 4 –2 The parent function is multiplied by a value l than 1 and greater than 0, so the the graph is a vertic compression of . Another way to identify the The parent function is multiplied by the negative of 1, so the graph is a reflection of across the x-axis. Another way to identify the graph i to notice that the y-values in the table are –1 times th corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}. 2. SOLUTION: Make a table. x 0 0.5 1 2 3 y 0 ≈ 0.18 0.25 ≈ 0.35 ≈ 0.43 Plot the points and draw a smooth curve. compression is to notice that the y-values in the table times the corresponding y-values for the parent functi The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}. 3. SOLUTION: Make a table. x 0 0.5 1 2 3 y 5 6 ≈ 5.7 ≈ 6.4 ≈ 6.7 Plot the points and draw a smooth curve. 4 7 The value 5 is being added to the parent function , so the graph is translated up 5 units from th The parent function is multiplied by a value l than 1 and greater than 0, so the the graph is a vertic eSolutions Manual - Powered by Cognero compression of . Another way to identify the compression is to notice that the y-values in the table parent graph . Another way to identify the translation is to note that the y-values in the table are greater than the corresponding y-values for the paren function. The domain is {x|x ≥ 0} and the range is {y| Page 1 ≥ 5}. compression of parent graph . Another way to identify the translation is to note that the y-values in the table are greater than the corresponding y-values for the paren function. The domain is {x|x ≥ 0} and the range is {y| ≥ 5}. . Another way to identify the compression is to notice that the y-values in the table timesTest the corresponding Practice - Chapter 10y-values for the parent functi The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}. 3. 4. SOLUTION: Make a table. x 0 0.5 1 2 3 y 5 6 ≈ 5.7 ≈ 6.4 ≈ 6.7 Plot the points and draw a smooth curve. SOLUTION: Make a table. x 0 2 4 –4 –2 y 0 2 ≈ 1.4 ≈ 2.4 ≈ 2.8 Plot the points and draw a smooth curve. 4 7 The value 5 is being added to the parent function , so the graph is translated up 5 units from th The value 4 is being added to the square root of the parent function , so the graph is translated 4 parent graph . Another way to identify the translation is to note that the y-values in the table are greater than the corresponding y-values for the paren function. The domain is {x|x ≥ 0} and the range is {y| ≥ 5}. units left from the parent graph . Another way to identify the translation is to note that the xvalues in the table are 4 less than the corresponding x-values for the parent function. The domain is {x|x ≥ –4}, and the range is {y|y ≥ 0}. 4. 5. GEOMETRY The length of the side of a square is SOLUTION: Make a table. x 0 2 4 –4 –2 y 0 2 ≈ 1.4 ≈ 2.4 ≈ 2.8 Plot the points and draw a smooth curve. given by the function s = , where A is the area of the square. What is the perimeter of a square that has an area of 64 square inches? A 64 inches B 8 inches C 32 inches D 16 inches SOLUTION: Substitute 64 for A. The side length of the square is 8 inches. The value 4 is being added to the square root of the parent function , so the graph is translated 4 units left from the parent graph . Another way to identify the translation is to note that the xvalues in the table are 4 less than the corresponding eSolutions Manual by Cognero x-values for- Powered the parent function. The domain is {x|x ≥ –4}, and the range is {y|y ≥ 0}. The perimeter of the square is 32 inches. The correct choice is C. Page 2 Simplify each expression. 6. units left from the parent graph . Another way to identify the translation is to note that the xvalues in the table are 4 less than the corresponding x-values the parent10 function. The domain is {x|x Practice Testfor- Chapter ≥ –4}, and the range is {y|y ≥ 0}. 5. GEOMETRY The length of the side of a square is given by the function s = , where A is the area of the square. What is the perimeter of a square that has an area of 64 square inches? A 64 inches B 8 inches C 32 inches D 16 inches 7. SOLUTION: SOLUTION: Substitute 64 for A. The side length of the square is 8 inches. 8. SOLUTION: The perimeter of the square is 32 inches. The correct choice is C. Simplify each expression. 6. SOLUTION: 7. 9. SOLUTION: 10. GEOMETRY Find the area of the rectangle. SOLUTION: F G 14 H eSolutions Manual - Powered by Cognero J SOLUTION: Page 3 The area of the rectangle is The correct choice is H. Practice Test - Chapter 10 Solve each equation. Check your solution. 10. GEOMETRY Find the area of the rectangle. square units. 11. SOLUTION: F G 14 Check: H J SOLUTION: Substitute for and for w. 12. SOLUTION: Check: The area of the rectangle is The correct choice is H. square units. Solve each equation. Check your solution. 11. SOLUTION: Because 13 does not satisfy the original equation, 3 is the only solution. 13. PACKAGING A cylindrical container of chocolate 3 drink mix has a volume of about 162 in . The radius of the container can be found by using the formula , where r is the radius and h is the height. Check: eSolutions Manual - Powered by Cognero If the height is 8.25 inches, find the radius of the container. SOLUTION: Substitute 8.25 for h. Page 4 A length cannot be negative. The missing length is 10 units. Because not satisfy Practice Test13- does Chapter 10 the original equation, 3 is the only solution. 13. PACKAGING A cylindrical container of chocolate 3 drink mix has a volume of about 162 in . The radius of the container can be found by using the formula , where r is the radius and h is the height. If the height is 8.25 inches, find the radius of the container. 15. SOLUTION: SOLUTION: Substitute 8.25 for h. A length cannot be negative. The missing length is approximately 9.2 units. The radius of the container is about 2.5 inches. Find each missing length. If necessary, round to the nearest tenth. 16. DELIVERY Ben and Amado are delivering a freezer. The bank in front of the house is the same height as the back of the truck. They set up their ramp as shown. What is the length of the slanted part of the ramp to the nearest foot? 14. SOLUTION: SOLUTION: Let L be the length of the slanted part of the ramp. L can be found using the Pythagorean theorem: A length cannot be negative. The missing length is 10 units. 17. Find the values of the three trigonometric ratios for angle A. 15. SOLUTION: SOLUTION: eSolutions Manual - Powered by Cognero Page 5 Keystrokes: [SIN] 9 15 The result is 36.86989764584. So, m X is about 37°. Practice Test - Chapter 10 17. Find the values of the three trigonometric ratios for angle A. 19. LIGHTHOUSE How tall is the lighthouse? SOLUTION: SOLUTION: Let h be the height of the lighthouse. L can be found using trigonometric ratios: 18. Find to the nearest degree. SOLUTION: You know the measure of the side opposite the angle and the hypotenuse. Use the sine ratio. -1 Use a calculator and the [sin ] function to find the measure of the angle. Keystrokes: [SIN] 9 15 The result is 36.86989764584. So, m X is about 37°. 19. LIGHTHOUSE How tall is the lighthouse? eSolutions Manual - Powered by Cognero Page 6 The roots of the equation are and . Choice A is the correct answer. Preparing for Standardized Tests - Chapter 9 Read each problem. Identify what you need to know. Then use the information in the problem to solve. 2 1. Find the exact roots of the quadratic equation x + 5x − 12 = 0. A 2. The area of a triangle in which the length of the base is 4 centimeters greater than twice the height is 80 square centimeters. What is the length of the base of the triangle? F −10 G 8 H 16 J 20 SOLUTION: You are asked to find the length of the base of a triangle. Use the formula for finding the area of a triangle. Let h = the height of the triangle and let 2h + 4 = the length of the base of the triangle. B C D SOLUTION: You are given a quadratic equation and asked to find the exact roots of the equation. Use the Quadratic Formula to find the roots. For this equation, a = 1, b = 5 and c = –12. The dimensions of the triangle cannot be negative. So, the height is 8 centimeters and the length of the base is 2(8) + 4 or 20 centimeters. Choice J is the correct answer. 3. Find the volume of the figure shown. The roots of the equation are and . Choice A is the correct answer. A 18.5 cm3 3 2. The area of a triangle in which the length of the base is 4 centimeters greater than twice the height is 80 square centimeters. What is the length of the base of the triangle? F −10 G 8 H 16 J 20 SOLUTION: You are asked to find the length of the base of a eSolutions Manual - Powered by Cognero triangle. Use the formula for finding the area of a triangle. Let h = the height of the triangle and let 2h + 4 = the length of the base of the triangle. B 91 cm C 272 cm3 D 292.5 cm 3 SOLUTION: You are asked to find the volume of a rectangular prism. Use the formula for the volume of a prism. Page 1 The volume of the rectangular prism is 292.5 cubic The dimensions of the triangle cannot be negative. So, the height is 8 centimeters and the length of the base is 2(8) +for 4 orStandardized 20 centimeters. Choice J is the 9correct Preparing Tests - Chapter answer. 3. Find the volume of the figure shown. A 18.5 cm3 3 B 91 cm C 272 cm3 D 292.5 cm 3 SOLUTION: You are asked to find the volume of a rectangular prism. Use the formula for the volume of a prism. The volume of the rectangular prism is 292.5 cubic centimeters. Choice D is the correct answer. 4. Myron is traveling 263.5 miles at an average rate of 62 miles per hour. How long will it take Myron to complete his trip? F 5h 25 min G 4h 15 min H 5h 10 min J 4h 25 min SOLUTION: You are asked to find the time it will take to travel a certain distance given the average rate. Use the formula for distance. It will take 4.25 hours or 4 hours and 15 minutes for Myron to complete his trip. Choice G is the correct answer. eSolutions Manual - Powered by Cognero Page 2 Standardized Test Practice - Cumulative, Chapter 1-10 1. Which equation below could match the graph shown on the coordinate grid? So, the graph of the equation in C will contain the points (1, 5) and (4, 7). The graph appears to contain the points (1, 1) and (4, –1). Therefore, the correct answer choice is B. 2. Simplify . F G A B C D SOLUTION: H J SOLUTION: From the graph, the y -intercept appears to be 3. The y -intercept of the graphs for the equations given in A and D are both 1. These two choices can be eliminated. Check the values of the points at x = 1 and 4 for the equations given in B and C and compare to the graph. Therefore, the correct answer is H. 3. What is the area of the triangle below? So, the graph of the equation in B will contain the points (1, 1) and (4, –1). A B C D SOLUTION: So, the graph of the equation in C will contain the points (1, 5) and (4,by7). eSolutions Manual - Powered Cognero The graph appears to contain the points (1, 1) and Page 1 Standardized Test Practice - Cumulative, Chapter 1-10 Therefore, the correct answer is H. 3. What is the area of the triangle below? Therefore, the correct answer is D. 4. The formula for the slant height c of a cone is , where h is the height of the cone and r is the radius of its base. What is the radius of the cone below? Round to the nearest tenth. A B C D SOLUTION: F 4.9 G 6.3 H 9.8 J 10.2 SOLUTION: Substitute h = 10 and c = 14 into the formula for the slant height of a cone to find the radius of the cone. Therefore, the correct answer is D. 4. The formula for the slant height c of a cone is , where h is the height of the cone and r is the radius of its base. What is the radius of the cone below? Round to the nearest tenth. The radius is about 9.8 units. Therefore, the correct answer is H. F 4.9 G 6.3 H 9.8 J 10.2 SOLUTION: Substitute h = 10 and c = 14 into the formula for the slant height of a cone to find the radius of the cone. 5. Which of the following sets of measures could not be the sides of a right triangle? A (12, 16, 24) B (10, 24, 26) C (24, 45, 51) D (18, 24, 30) SOLUTION: A Since the measure of the longest side is 24, let c = 2 24, a = 12, and b = 16. Then determine whether c = 2 2 a +b . eSolutions Manual - Powered by Cognero Page 2 SOLUTION: A Since the measure of the longest side is 24, let c = Standardized Test Practice - Cumulative, Chapter2 1-10 24, a = 12, and b = 16. Then determine whether c = 2 2 a +b . Because 900 = 900, a triangle with side lengths 18, 24, and 30 is a right triangle. Therefore, the correct answer is A. 6. Which of the following is an equation of the line perpendicular to 4x – 2y = 6 and passing through (4, –4)? F Because 576 ≠ 400, a triangle with side lengths 12, 16, and 24 is not a right triangle. B Since the measure of the longest side is 26, let c = 2 26, a = 10, and b = 24. Then determine whether c = 2 2 a +b . G H J SOLUTION: First, find the slope of 4x – 2y = 6 by writing the equation in slope-intercept form. Because 676 = 676, a triangle with side lengths 10, 24, and 26 is a right triangle. C Since the measure of the longest side is 51, let c = 2 51, a = 24, and b = 45. Then determine whether c = 2 2 a +b . So, the slope is 2, and a line perpendicular to y = 2x – 3 will have a slope of . Next, find the line passing through (–4, 4). Because 2601 = 2601, a triangle with side lengths 24, 45, and 51 is a right triangle. D Since the measure of the longest side is 30, let c = 2 30, a = 18, and b = 24. Then determine whether c = 2 2 a +b . Because 900 = 900, a triangle with side lengths 18, 24, and 30 is a right triangle. Therefore, the correct answer is A. eSolutions Manual - Powered by Cognero 6. Which of the following is an equation of the line perpendicular to 4x – 2y = 6 and passing through (4, Therefore, the correct answer is J. 7. The scale on a map shows that 1.5 centimeters is equivalent to 40 miles. If the distance on the map between two cities is 8 centimeters, about how many miles apart are the cities? A 178 miles B 213 miles C 224 miles D 275 miles SOLUTION: Use ratios to find the distance. Page 3 SOLUTION: Standardized Test Practice - Cumulative, Chapter 1-10 Therefore, the correct answer is J. 7. The scale on a map shows that 1.5 centimeters is equivalent to 40 miles. If the distance on the map between two cities is 8 centimeters, about how many miles apart are the cities? A 178 miles B 213 miles C 224 miles D 275 miles SOLUTION: Use ratios to find the distance. 10. GRIDDED RESPONSE In football, a field goal is worth 3 points, and the extra point after a touchdown is worth 1 point. During the 2006 season, John Kasay of the Carolina Panthers scored a total of 100 points for his team by making a total of 52 field goals and extra points. How many field goals did he make? SOLUTION: Setup and solve a system in equations where x represents the number of field goals and y represents the number of extra points. First, solve for x. The cities are about 213 miles apart. Therefore, the correct answer is B. 8. GRIDDED RESPONSE How many times does the 2 graph of y = x - 4x + 10 cross the x-axis? SOLUTION: 2 Since the equation y = x – 4x + 10 is a quadratic, it will cross the x-axis 0, 1, or 2 times. Graph the equation. Therefore, the number of field goals is 24. 11. Shannon bought a satellite radio and a subscription to satellite radio. What is the total cost for his first year of service? SOLUTION: Let C represent the total cost for the first year of service. [–10, 10] scl: 1 by [–10, 10] scl: 1 2 The graph of y = x – 4x + 10 does not cross the xaxis. Therefore, the answer is 0. 9. Factor completely. SOLUTION: Therefore, the total cost is $183.87. 12. GRIDDED RESPONSE The distance required for a car to stop is directly proportional to the square of its velocity. If a car can stop in 242 meters at 22 kilometers per hour, how many meters are needed to stop at 30 kilometers per hour? SOLUTION: Substitute 242 for d and 22 for v. 10. GRIDDED RESPONSE In football, a field goal is worth 3 points, and the extra point after a touchdown eSolutions Manual - Powered by Cognero is worth 1 point. During the 2006 season, John Kasay of the Carolina Panthers scored a total of 100 points for his team by making a total of 52 field goals and Page 4 Standardized Test Practice - Cumulative, Chapter 1-10 Therefore, the total cost is $183.87. 12. GRIDDED RESPONSE The distance required for a car to stop is directly proportional to the square of its velocity. If a car can stop in 242 meters at 22 kilometers per hour, how many meters are needed to stop at 30 kilometers per hour? SOLUTION: Substitute 242 for d and 22 for v. 15. GRIDDED RESPONSE For the first home basketball game, 652 tickets were sold for a total revenue of $5216. If each ticket costs the same, how much is the cost per ticket? State your answer in dollars. SOLUTION: Let x be the cost of a ticket. Thus, each ticket costs $8.00. A car will need 450 meters to stop at 30 kilometers per hour. 13. The highest point in Kentucky is at an elevation of 4145 feet above sea level. The lowest point in the state is at an elevation of 257 feet above sea level. Write an inequality for this situation. 16. Karen is making a map of her hometown using a coordinate grid. The scale of her map is 1 unit = 2.5 miles. SOLUTION: 257 ≤ x ≤ 4,145 14. Simplify the expression below. Show your work. SOLUTION: a. Use the Pythagorean Theorem to find the actual distance between Karen’s school and the park. Round to the nearest tenth of a mile if necessary. b. Suppose Karen’s house is located at (0.5, 0.5). What is farthest from her house, the zoo, the park, the school, or the mall? 15. GRIDDED RESPONSE For the first home basketball game, 652 tickets were sold for a total revenue of $5216. If each ticket costs the same, how much is the cost per ticket? State your answer in dollars. SOLUTION: Let x be the cost of a ticket. eSolutions Manual - Powered by Cognero SOLUTION: a. The coordinates of the school are (4, 5), and the coordinates of the park are (5, –5). Use the Pythagorean theorem to find the distance between the school and the park. Page 5 SOLUTION: a. The coordinates of the school are (4, 5), and the coordinates of the park are (5, –5). Use the Pythagorean theorem to find the distanceChapter between 1-10 Standardized Test Practice - Cumulative, the school and the park. The school is located at (4, 5), so the distance between Karen's house and the school is: So, the distance is 10.04 units. Since 1 unit = 2.5 miles, the distance in miles is 10.04 · 2.5 or about 25.1 miles. The park is 7.11 units away from Karen's house, the school is 5.7, and the zoo is 4.75. The park is farthest from Karen's house, which will be true even if the scale is changed. b. Let (x 1, y 1) = (0.5, 0.5). The zoo is located at (– 4, 2), so the distance between Karen's house and the zoo is: The park is located at (5, –5), so the distance between Karen's house and the park is: The school is located at (4, 5), so the distance between Karen's house and the school is: eSolutions Manual - Powered by Cognero Page 6 SOLUTION: In a right triangle, the side opposite the right angle is the hypotenuse. This side is always the longest. So, the statement is false. The longest side of a right triangle is the hypotenuse. Study Guide and Review - Chapter 10 State whether each sentence is true or false . If false , replace the underlined word, phrase, expression, or number to make a true sentence. 1. A triangle with sides having measures of 3, 4, and 6 is a right triangle. SOLUTION: 2 If a, b, and c are the sides of a right triangle, then c 2 2 2 = a + b , where c is the greatest number. Since 6 2 2 2 2 2 = 36 and 3 + 4 = 25, 6 ≠ 3 + 4 . Thus, a triangle with sides having measures of 3, 4, and 6 is not a 2 2 right triangle. So, the statement is false. Since 3 + 4 2 = 25 and 5 = 25, a triangle with sides having measures of 3, 4, and 5 would be a right triangle. 2. The expressions and are equivalent. 6. The cosine of an angle is found by dividing the measure of the side opposite to the angle by the hypotenuse. SOLUTION: The sine of an angle is found by dividing the measure of the side opposite of the angle by the hypotenuse. So, the statement is false. The cosine of an angle is found by dividing the measure of the side adjacent to the angle by the hypotenuse. 7. The domain of the function is . SOLUTION: The domain of the function the statement is false. is {x|x ≥ 0}. So, 8. After the first step in solving = x + 5, you 2 would have 2x + 4 = x + 10x + 25. SOLUTION: SOLUTION: false; To solve = x + 5, you first need to square each side of the equation. After the first step in 3. The expressions 2 + and are conjugates. SOLUTION: Binomials of the form and are called conjugates. So, the binomials and are conjugates. The statement is true. 4. In the expression −5 solving = x + 5, you would have 2x + 4 = x + 10x + 25. The statement is true. 2 9. The converse of the Pythagorean Theorem is true. SOLUTION: 2 2 2 If c ≠ a + b , then the triangle is not a right triangle. So, the converse of the Pythagorean Theorem is true. The statement is true. , the radicand is 2. SOLUTION: The expression under the radical sign is called the radicand. In the expression , the radicand is 2. The statement is true. 5. The shortest side of a right triangle is the hypotenuse. SOLUTION: In a right triangle, the side opposite the right angle is the hypotenuse. This side is always the longest. So, the statement is false. The longest side of a right triangle is the hypotenuse. 6. The cosine of an angle is found by dividing the measure of the side opposite to the angle by the hypotenuse. SOLUTION: The Manual sine of- an anglebyisCognero found by dividing the measure eSolutions Powered of the side opposite of the angle by the hypotenuse. So, the statement is false. The cosine of an angle is found by dividing the measure of the side adjacent 10. The range of the function is . SOLUTION: is {y|y ≥ 0}. So, The range of the function the statement is false. Graph each function. Compare to the parent graph. State the domain and range. 11. y = – 3 SOLUTION: Make a table. x y 0 3 0.5 ≈ –2.3 1 –2 2 ≈ – 1.6 3 ≈ –1.3 Page 1 Plot the points on a coordinate systems and draw a curve that connects them. 10. The range of the function is . SOLUTION: The rangeand of the function Study Guide Review - Chapter is {y|y ≥ 0}. So, 10 the statement is false. Graph each function. Compare to the parent graph. State the domain and range. 11. y = Make a table. 0 3 12. y = + 2 SOLUTION: Make a table. x 0 0.5 y 2 ≈ 2.7 – 3 SOLUTION: x y parent graph . Another way to identify the translation is to note that the y-values in the table are than the corresponding y-values for the parent functi domain is {x|x ≥ 0} and the range is {y|y ≥ –3}. 0.5 ≈ –2.3 1 –2 2 ≈ – 1.6 3 ≈ –1.3 1 3 2 ≈ 3.4 3 ≈ 3.7 4 4 Plot the points on a coordinate system and draw a smooth curve that connects then. Plot the points on a coordinate systems and draw a curve that connects them. The value 2 is being added to the parent function , so the graph is translated up 2 units from The value 3 is being subtracted from the parent funct , so the graph is translated down 3 units from parent graph . Another way to identify the translation is to note that the y-values in the table are than the corresponding y-values for the parent functi domain is {x|x ≥ 0} and the range is {y|y ≥ –3}. 12. y = 13. y = −5 SOLUTION: x 0 y 0 + 2 SOLUTION: Make a table. x 0 0.5 y 2 ≈ 2.7 the parent graph . Another way to identify the translation is to note that the y-values in the table are 2 greater than the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 2}. 1 3 2 ≈ 3.4 3 ≈ 3.7 0.5 ≈ –3.5 1 –5 2 ≈ –7.1 4 4 Plot the points on a coordinate system and draw a smooth curve that connects then. The parent function is multiplied by a value less than 1, so the graph is a vertical stretch of and a reflection across the x-axis. Another way to identify the stretch is to notice that the yvalues in the table are –5 times the corresponding yvalues for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}. eSolutions Manual - Powered by Cognero Page 2 14. y = The value 2 is being added to the parent function − 6 SOLUTION: the parent graph . Another way to identify the translation is to note that the y-values in the table are 2 greater than the corresponding y-values for the Study Guide and Review - Chapter 10 parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 2}. 13. y = −5 15. y = SOLUTION: x 0 y 0 0.5 ≈ –3.5 1 –5 2 ≈ –7.1 The parent function is multiplied by a value less than 1, so the graph is a vertical stretch of and a reflection across the x-axis. Another way to identify the stretch is to notice that the yvalues in the table are –5 times the corresponding yvalues for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}. 14. y = units from the parent graph . Another way to identify the translation is to note that the y-values in the table are 6 less than the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ –6}. − 6 SOLUTION: x 0 0.5 y –6 ≈ – 5.3 1 –5 2 ≈ – 4.6 3 ≈ – 4.3 4 –4 units from the parent graph . Another way to identify the translation is to note that the y-values in the table are 6 less than the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ –6}. 15. y = 2 1 eSolutions Manual - Powered by Cognero 3 ≈ 1.4 4 ≈ 1.7 2 1 3 ≈ 1.4 4 ≈ 1.7 The value 1 is being subtracted from the square root of the parent function , so the graph is translated 1 unit right from the parent graph . Another way to identify the translation is to note that the x-values in the table are 1 more than the corresponding x-values for the parent function. The domain is {x|x ≥ 1}, and the range is {y|y ≥ 0}. 16. y = The value 6 is being subtracted from the parent function , so the graph is translated down 6 SOLUTION: x 1 1.5 y 0 ≈ 0.7 SOLUTION: x 1 1.5 y 0 ≈ 0.7 + 5 SOLUTION: x 0 0.5 y 5 ≈ 5.7 1 6 2 ≈ 6.4 3 ≈ 6.7 4 7 The value 5 is being added to the parent function , so the graph is translated up 5 units from the parent graph . Another way to identify the translation is to note that the y-values in the table are 5 greater than the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 5}. 17. GEOMETRY The function s = can be used to find the length of a side of a square given its area. Use this function to determine the length of a side of a square with an area of 90 square inches. Round to the nearest tenth if necessary. SOLUTION: Page 3 the parent graph . Another way to identify the translation is to note that the y-values in the table are 5 greater than the corresponding y-values for the Study Guide and Review - Chapter 10 parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 5}. 17. GEOMETRY The function s = can be used to find the length of a side of a square given its area. Use this function to determine the length of a side of a square with an area of 90 square inches. Round to the nearest tenth if necessary. 22. SOLUTION: SOLUTION: 23. SOLUTION: The side length of the square is about 9.5 inches. Simplify. 18. SOLUTION: 24. SOLUTION: 19. SOLUTION: 20. SOLUTION: 25. SOLUTION: 21. SOLUTION: 22. eSolutions Manual - Powered by Cognero SOLUTION: Page 4 Study Guide and Review - Chapter 10 27. 25. SOLUTION: SOLUTION: 28. WEATHER To estimate how long a thunderstorm will last, use , where t is the time in hours and d is the diameter of the storm in miles. A storm is 10 miles in diameter. How long will it last? SOLUTION: Substitution 10 for d. 26. SOLUTION: The storm will last about 2.15 hours. To convert 2.15 hours to hours and minutes, multiply the number of minutes in an hour by the decimal part. Because 60 • 0.15 = 9, 2.15 hours is equal to 2 hours and 9 minutes. Simplify each expression. 29. SOLUTION: 27. SOLUTION: 30. SOLUTION: eSolutions Manual - Powered by Cognero Page 5 SOLUTION: Study Guide and Review - Chapter 10 36. MOTION The velocity of a dropped object when it 30. hits the ground can be found using , where v is the velocity in feet per second, g is the acceleration due to gravity, and d is the distance in feet the object drops. Find the speed of a penny when it hits the ground, after being dropped from 984 feet. Use 32 feet per second squared for g. SOLUTION: SOLUTION: Substitute 32 for g and 984 for d. 31. SOLUTION: 32. SOLUTION: The speed of the penny when it hits the ground is about 250.95 feet per second. Solve each equation. Check your solution. 33. 37. SOLUTION: SOLUTION: 34. SOLUTION: Because the square root of a number cannot be negative, there is no solution. 38. SOLUTION: 35. SOLUTION: 36. MOTION The velocity of a dropped object when it hits the ground can be found using , where v is the velocity in feet per second, g is the acceleration due to gravity, and d is the distance in feet the object drops. Find the speed of a penny whenManual it hits- Powered the ground, after being dropped from 984 eSolutions by Cognero feet. Use 32 feet per second squared for g. SOLUTION: Check: Page 6 Because the square root of a number cannot be Study Guide there and Review - Chapter 10 negative, is no solution. 38. 40. SOLUTION: SOLUTION: Check: Check: 41. SOLUTION: 39. SOLUTION: Check: Check: Because 5 does not satisfy the original equation, 12 is the only solution. 40. SOLUTION: 42. SOLUTION: eSolutions Manual - Powered by Cognero Check: Page 7 Check: Because doesReview not satisfy the original Study Guide5and - Chapter 10 equation, 12 is the only solution. The skydiver will fall 1600 feet before opening the parachute. Determine whether each set of measures can be lengths of the sides of a right triangle. 44. 6, 8, 10 42. SOLUTION: SOLUTION: Since the measure of the longest side is 10, let c = 2 10, a = 6, and b = 8. Then determine whether c = 2 2 a +b . Check: 2 2 2 Yes, because c = a + b , a triangle with side lengths 6, 8, and 10 is a right triangle. 45. 3, 4, 5 Because 10 does not satisfy the original equation, 5 is the only solution. 43. FREE FALL Assuming no air resistance, the time t in seconds that it takes an object to fall h feet can be determined by SOLUTION: Since the measure of the longest side is 5, let c = 5, a 2 2 = 3, and b = 4. Then determine whether c = a + 2 b . . If a skydiver jumps from an airplane and free falls for 10 seconds before opening the parachute, how many feet does she fall? SOLUTION: Substitute 10 for t. 2 2 2 Yes, because c = a + b , a triangle with side lengths 3, 4, and 5 is a right triangle. 46. 12, 16, 21 SOLUTION: Since the measure of the longest side is 21, let c = 2 21, a = 12, and b = 16. Then determine whether c = 2 2 a +b . 2 The skydiver will fall 1600 feet before opening the parachute. Determine whether each set of measures can right triangle. 44. 6, 8, 10 eSolutions Manual - Powered by Cognero be lengths of the sides of a SOLUTION: 2 2 No, because c ≠ a + b , a triangle with side lengths 12, 16, and 21 is not a right triangle. 47. 10, 12, 15 SOLUTION: Since the measure of the longest side is 15, let c Page = 8 2 15, a = 10, and b = 12. Then determine whether c = 2 2 a +b . 2 2 2 No, because , a triangle Study Guide andc Review 10with side lengths ≠ a + -bChapter 12, 16, and 21 is not a right triangle. 47. 10, 12, 15 2 a +b . 2 2 2 No, because c ≠ a + b , a triangle with side lengths 10, 12, and 15 is not a right triangle. 48. 2, 3, 4 2 SOLUTION: Since the measure of the longest side is 13, let c = 2 13, a = 5, and b = 12. Then determine whether c = 2 2 a +b . 2 2 2 Yes, because c = a + b , a triangle with side lengths 5, 12, and 13 is a right triangle. 51. 15, 19, 23 SOLUTION: Since the measure of the longest side is 4, let c = 4, a 2 2 = 2, and b = 3. Then determine whether c = a + 2 b . 2 2 2 No, because c ≠ a + b , a triangle with side lengths 2, 3, and 4 is not a right triangle. 49. 7, 24, 25 SOLUTION: Since the measure of the longest side is 25, let c = 2 25, a = 7, and b = 24. Then determine whether c = 2 2 50. 5, 12, 13 SOLUTION: Since the measure of the longest side is 15, let c = 2 15, a = 10, and b = 12. Then determine whether c = 2 2 Yes, because c = a + b , a triangle with side lengths 7, 24, and 25 is a right triangle. 2 a +b . 2 2 SOLUTION: Since the measure of the longest side is 23, let c = 2 23, a = 15, and b = 19. Then determine whether c = 2 2 a +b . 2 2 2 No, because c ≠ a + b , a triangle with side lengths 15, 19, and 23 is not a right triangle. 52. LADDER A ladder is leaning on a building. The base of the ladder is 10 feet from the building, and the ladder reaches up 15 feet on the building. How long is the ladder? SOLUTION: Use the Pythagorean Theorem, substituting 10 for a and 15 for b. 2 Yes, because c = a + b , a triangle with side lengths 7, 24, and 25 is a right triangle. 50. 5, 12, 13 The ladder is approximately 18.0 feet long. SOLUTION: Since the measure of the longest side is 13, let c = 2 13, a = 5, and b = 12. Then determine whether c = 2 2 eSolutions a +Manual b . - Powered by Cognero Find the values of the three trigonometric ratios for angle A . Page 9 Study Guide and Review - Chapter 10 The ladder is approximately 18.0 feet long. Find the values of the three trigonometric ratios for angle A . 55. RAMPS How long is the ramp? 53. SOLUTION: SOLUTION: You know the measure of the side opposite the angle and the measure of the angle. Use the sine ratio. The ramp is 6 feet long. 54. SOLUTION: 55. RAMPS How long is the ramp? eSolutions Manual - Powered by Cognero SOLUTION: You know the measure of the side opposite the angle Page 10