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Essentials of Math 11 Booklet #4 – Slope & Rate of Change Instructor – Paula Nelson ACC Adult Collegiate Fall 2012 1 Table of Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. Learning Outcomes……………………………….. Terms to Know…………………………………… Review of Prior Knowledge………………………. Lesson 1: Rise Over Run Lesson 2: Lesson 3: Hand In Assignment……………………………… Test Overview & Review………………………… Trigonometry Review……………………………. Page 3 Pages 3 Pages 4-5 Pages 6-8 Pages 8-10 Pages 10-14 Page 15 Page 15 Page 16-17 Assignments: Lesson Number 1. 2. 3. Topic Assignment Mark Value Terms Define the Terms Listed on page 4 (use your text) 10 Review Questions Rise over Run Grade, Angle of Elevation and Distance Rate of Change Page 19 #1-10 Page 30 #1-6 Review Test Formula sheet 2 - Page 46 #1-9 Hand In Assignment 15 Page 52 #1-8 10 50 5 Learning Outcomes: By the end of this unit is it expected that students will be able to: Demonstrate an understanding of slope: o as rise over run o as rate of change by solving problems. Solve problems by applying proportional reasoning and unit analysis. Solve problems that require the manipulation and application of formulas related to slope and rate of change. Solve problems that involve scale. Demonstrate an understanding of linear relations by: o recognizing patterns and trends o graphing o creating tables of values o writing equations o interpolating and extrapolating o solving problems. DEFINE THESE TERMS ON A SEPARATE SHEET OF PAPER TERMS to KNOW: see text Chapter 7 for definitions Rise Run Grade Tangent ratio Angle of elevation Pythagorean Theorem Angle of Depression Drop Rate of Change Dependent Variable Independent Variable Zero Slope Undefined Slope 3 Review 4 5 Lesson 1: Rise Over Run Slope is a numerical value that tells you how steep something is or how slanted it is. We can calculate how steep something is with a simple formula. Other names for slope include pitch, gradient or incline. Slope is calculated my measuring the amount of vertical change in distance (called rise) and comparing it (or dividing it) by the change in horizontal distance (called run). This comparison, or ratio, can be expressed as follows: Slope = ∆ vertical distance ∆ hortizontal distance In math, we use m to represent slope (this is our symbol for slope). So, therefore… m = rise run 6 Example 1 Charlene is working on a major project in her Art Metal class. She wants to make a brass cookbook holder like the one in the diagram. She does not have a protractor to measure the angle, but the diagram is drawn on graph paper. a) What is the slope of the cookbook holder? b) What does your answer tell you about the relationship between vertical change and horizontal change of the cookbook holder? c) Andy says the slope is 3/2. Amanda uses her calculator and gets the answer 1.5. Jennifer says the slope is 6:4. Explain who is correct. Solution a) Determine the rise and the run by counting the number of squares on the grid. The rise is 6 units and the run is 4 units. m = 6/4 m = 3/2 substitute the rise and run values into the slope formula simplify the fraction The slope of the cookbook holder is 3/2. b) The slope is a comparison of vertical change (rise) over the horizontal change (run). In this example, the slope of the cookbook holder changes vertically a distance of 3 units for every 2 units it changes horizontally. c) They are all correct – in order to compare and verify the answer, reduce all representations to a decimal (3/2 = 1.5; 1.5 is already in decimal form; 6:4 is 6 divided by 4 which is 1.5). Example 2 Duncan and Casey are using hand trucks to move small boxes from a house to a garage. They lay a loading ramp against the house step, which are 18 inches high. The slope of the ramp is 0.2. What is the horizontal distance (run), in feet, from the base of the ramp to point x? ** note – there are 12 inches in 1 foot. Solution Substitute the know values into the slope formula. 0.2 = 18 0.2(run) = 18 run = 18 run 0.2 7 run = 90” 90/12 = 7.5 ft ASSIGNMENT: Read pages 12-19 (leave out Activities) and follow the examples carefully Do Building your Skills questions page 19 #1-10 Lesson 2: Grade, Angle of Elevation and Distance If we are talking about the slope of a road – we call is the grade of the surface (road). Grade is generally expressed as a percentage – the steeper the slope, the higher the percent grade. To convert slope to percent grade, multiply the numerical value for slope by 100. The rise and run must be the in the same units (i.e. both in metres) when calculating percent grade. Percent grade = rise run x 100 The slope of the line (rise/run) is the tangent (opposite/adjacent) of the angle of elevation. When you know the tangent ratio (slope), you can use the inverse tangent function [tan-1 (x)] to find the angle of elevation. The slope is equal to the tangent of the angle of elevation, so you can convert an angle of elevation to a percent grade by using the following expression: tan θ x 100 = percent grade In addition to calculating the steepness of a slope, you will often need to calculate the length of a line that represents a particular slope. Use can use the Pythagorean Theorum to calculate the length of the hypotenuse. The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle. A right triangle is any triangle that has one right (90◦) internal angle. Pythagoras stated, if the length of the legs (smallest side) are squared and their sum is found, the sum will be equal to the square of the hypotenuse (longest side). Algebraically speaking, the relationship looks like... 8 The legs are traditionally marked with an 'a' and a 'b,' while the hypotenuse is marked with a 'c.' If that is the case, then the formula is a2 + b2 = c2. The next section will explain how the equation can be derived. Rearranged: c = √a2 + b2 a = √c2 - b2 b = √c2 + a2 Example 1 Brad needs to unload a quad from the box of his pickup truck. He places an aluminum ramp against the truck bed at a slope of 7:40. What is the angle of elevation of the ramp? Angle of elevation is the angle formed by a horizontal line segment and an inclined plane segment – the angle between the horizontal and the object looking up. Solution Tan Ө = opposite side Tan Ө = rise Adjacent side run Tan Ө = 7 sub the known values in 40 Tan Ө = 0.175 convert the fraction to a decimal Ө = tan-1 (0.175) use the inverse tangent key on your calculator Ө = 9.9◦ the angle of elevation of the ramp is 9.9◦ Example 2 The slope of a driveway must have a minimum angle of depression of 1◦ to allow surface water to drain away from the house. If the end of a driveway is 8m from the house, how many centimetres does the driveway need to drop in order to maintain proper drainage? Round your answer to two decimal places. Solution Use the tangent formula to find the drop (rise) of the driveway. Tan Ө = rise Run ◦ Tan 1 = rise substitute known values into the formula 8 ◦ (8) tan 1 = rise plug into your calculator 0.1396 ~ rise 0.14 m ~ rise to convert to centimetres, multiply by 100 = 14 cm 9 Example 3 Josette wants to build a skateboard ramp with a 20% grade so that the top of the ramp is level with a rail that is 30 cm high. How long does the ramp need to be? Round your answer to the nearest centimetre. Solution First, calculate the slope of a 20% grade. Since grade is another word for slope substitute 20 in for m. 20 = rise x 100 the 100 is in there to convert the 20 into a percent Run 20/100 = rise/run 1/5 = rise/run a 20% grade has a slope of 1:5 The rise of Josette’s ramp is 30 cm, so sub this value into the formula and solve for (calculate) the run. 1 = 30 cross multiply and divide 5 x 30 = 1 x run 150 = run run=150cm 5 run Now to solve for the length of the ramp, you must put it into Pythagoras and solve for “c” c = √a2 + b2 c = √rise2 + run2 c = √302 + 1502 c = √900 + 22500 c = √23400 length or c = 152.97 cm The ramp needs to be about 153 cm long. ASSIGNMENT: Read pages 25-30 and follow the examples carefully Do Building your Skills questions page 30 #1-6 10 Lesson 3: Rate of Change http://www.regentsprep.org/Regents/math/ALGEBRA/AC1/Rate.h tm 11 A rate of change is the rate at which on variable changes compared to another variable. Therefore, one is known as the dependent variable and the other is known as the independent variable. You are required to be able to tell me the difference and some students have a lot of trouble with this. The dependent variable is a variable whose value relies on the values of the other variable (i.e. if you are planning a wedding the cost of the food relies on the number of guests, so cost the cost depends on the number of guests). The independent variable is a variable whose values is freely chosen (i.e. often time is independent or in the example of above, the number of guests at the wedding). When the relationship between the variables is constant, it has a linear relationship (makes a straight line on a graph). Remember – a horizontal line has a zero slope and a vertical line has an undefined slope. Slope can also be positive or negative depending on which way the line is rising or falling (see pics of ski bird above). Example 1 Kalirraq and Uyarak drive delivery trucks. Kalirraq gets paid $20.00 per hour and Uyarak gets paid $16.00 an hour plus a $20 gas allowance at the beginning of each workday for using his own vehicle. (a) Write an equation you can use to calculate each person’s earnings, using p for pay and h for hours. Graph 5 points of data for each person on the same graph. (b) Who makes more money after 3 hours of work? How much more? (c) When will they make the same amount of money? (d) Who makes more money after 9 hours of work? How much more? (e) Find the slope of the line segments that represent Kalirraq’s and Uyarak’s pay for hours worked. What can you conclude about these values? Solution (a) Make a data table for each person that includes five points. Kalirraq Uyarak Hours (h) Pay (p) Hours (h) 1 20 1 2 40 2 4 80 4 6 120 6 8 160 8 Pay (p) 20+16 = 36 36+16 = 52 52 + 84 116 148 Write an EQUATION FOR EACH PERSON: Kalirraq: pay = hours worked times $20/hour so an equation would be: p = 20(h) Uyarak: pay = hours worked times $16/hour plus $20 so the equation would be p = 16(h) +20 Now draw a graph that compares the dependent variable (pay) on the y axis to the independent variable (hours) on the x-axis 12 (independent variable is always graphed on the x axis). Plot both sets of points on the same graph. (b) Find the point on the graph when Kalirraq’s line segment matches the 3 hour value on the x-axis and match it to the corresponding value on the y axis. Kalirraq earns $60. DO the same for Uyarak’s line; he earns $68.00 $68-60 = $8 after 3 hours of work, Uyarak earns $8 more. (c) The two lines intersect (cross) at 5 hours, so they are making the same amount of money when they have worked 5 hours. (d) To find how much money they make after 9 hours, extrapolate by extending the line and finding the point where the line segments cross the 9 hour value. Kalirraq is making $180 and Uyarak is making $164 – Kalirraq makes $16 more. (e) Kalirraq: m = 120/6 m = 20 the slope is 20 Uyarak: 148 – 20 = 128 m = 128/8 m = 16 the slope is 16 The line representing Kalirraq’s pay is rising more steeply than Uyarak’s. The rate of change is greater for Kalirraq. Example 2 13 ASSIGNMENT: Read pages 35-46 and follow the examples carefully Do Building your Skills questions page 46 #1-9 More Practice on Writing Equations http://www.math123xyz.com/Nav/Algebra/Writing%20Linear%20Equations_Tutorial.ph p 14 /30 Hand In Assignment: Do Activity 1.7 SLOPE DIRECTION on page 40 of your Text 23 marks for answering/completing questions 1-10 5 marks for graphs 2 marks for neatness (ruler, colour…) and presentation Test Overview: tba Mark breakdown /50 o _______________________________________ o _______________________________________ o _______________________________________ o _______________________________________ YOU ARE ABLE TO HAVE A 8.5 x 11 CHEAT SHEET made up ahead of time (5 marks) You should be able to: o _________________________________________________________ o _________________________________________________________ o _________________________________________________________ o _________________________________________________________ o _________________________________________________________ o _________________________________________________________ o _________________________________________________________ o _________________________________________________________ o _________________________________________________________ o _________________________________________________________ REVIEW – Practice your New Skills text page 52 #1-8 15 Trigonometry Review Let's start with a quick review of the three trigonometric functions we already understand. Remember that these functions work only in right triangles. where A represents the angle of reference. Example: In right triangle ABC, hypotenuse length AB=15 and angle A=35º. Find leg length BC to the nearest tenth. Set up the problem: Draw a right triangle. Position yourself at the given angle as a point of reference. Label the triangle's sides as to their location: opposite, hypotenuse, adjacent. Place the given information on the triangle. Pair up the sides' labels with the given information. The piece with no pairing is ignored. Set up the function based upon the pairings and solve. Nothing is paired with side a in this problem, so it is ignored. This problem deals with o and h which means we will be working with the sin function. ANSWER: 8.6 Example: In right triangle ABC, leg length BC=15 and leg length AC=20. Find angle A to the nearest degree. 16 Set up the problem: The set up is the same as in the previous example. The only difference is that in this problem you will be finding the angle. Nothing is paired with side h in this problem, so it is ignored. This problem deals with o and a which means we will be working with the tan function. Calculator use: Remember you will need to activate the tan-1 key (it is located above the tan key) when finding angles. To activate this tan-1 key on most scientific calculators, enter 0.75, press 2nd (or shift) and then the tan key. On the graphing calculator, activate the tan-1 first, and then enter 0.75. 17