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April 15/16, 2009 MPM2D1 Name: ______________________ Maximum & Minimum – Day 2 – “Numbers” Example 4 The sum of two numbers is 60. Find the numbers if their product is a maximum. Our 2 numbers are n & 60-n. So, our formula is: p n(60 n) , where n is one We need to write formula with one variable to represent the problem. number and p is the product. p n(60 n) p n 2 60n Put the formula into standard form. Now we are ready to COMPLETE THE SQUARE! p (n 2 60n) p (n 2 60n 900 900) p (n 2 60n 900) 900 Factor out the a value. Calculate and include the magic number! 60 22 = 900 Boot the negative out of the brackets (remember to multiply by the negative in front of the brackets). Rewrite the perfect square portion in its factored p (n 30) 900 form. Simplify if necessary. Translate the information to make sense of the Vertex: (30, 900) question. At the maximum, the value of n is 30. If one number is 30, the other number is 60-n, which evaluates to 30 as well. So, the two numbers are 30 and 30. 2 Example 5 The sum of two numbers is 28. Find the numbers if the sum of their squares is a minimum. Our 2 numbers are n & 28-n. So, our formula is: minimum= n 2 (28 n) 2 , where n is one number and 28-n is the other number. minimum= n 2 784 56n n 2 = 2n 2 56n 784 We need to write formula with one variable to represent the problem. Put the formula into standard form. Now we are ready to COMPLETE THE SQUARE! minimum= 2(n 2 28n) 784 minimum= 2(n 2 28n 196 196) 784 minimum= 2(n 28n 196) 392 784 2 Factor out the a value. Calculate and include the magic number! 28 22 = 196 Boot the negative out of the brackets (remember to multiply by the negative in front of the brackets). April 15/16, 2009 MPM2D1 Name: ______________________ minimum= 2(n 14) 2 392 Vertex: (14, 392) Therefore, the two numbers are 14 and 14. Rewrite the perfect square portion in its factored form. Simplify if necessary. Translate the information to make sense of the question. Example 6 Two numbers have a difference of 16. Find the numbers if the result of adding their sum and their product is a minimum. sum Our 2 numbers are n & 16+n. So, our formula is: minimum= [n (16 n)] [n(16 n)] , where n is one number and 16+n is the other number. minimum= [n (16 n)] [n(16 n)] 2n 16 n 2 16n product We need to write formula with one variable to represent the problem. Put the formula into standard form. n 18n 16 2 Now we are ready to COMPLETE THE SQUARE! minimum= n 2 18n 16 minimum= n 2 18n 81 81 16 minimum= (n 9) 2 65 Vertex: (-9, -65) Therefore, our two numbers are -9 and 7. Factor out the a value. (BUT THERE IS NONE) Calculate and include the magic number! 18 22 = 81 Nothing to boot out of brackets! Rewrite the perfect square portion in its factored form. Simplify if necessary. Translate the information to make sense of the question. Example 7 The sum of a number and three times another number is 18. Find the numbers if their product is a maximum. Assignment: 1. Page 272 #18, 20, 22 (Day1 type questions) AND 2. The sum of two natural numbers is 12. If their product is a maximum, find the numbers. 3. Two numbers have a difference of 20. Find the numbers if the sum of their squares is a minimum.