Richland Community College The Inverse and Direct Variation Problems

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Richland Community College

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Inverse Variation The flowchart below shows how to decide whether a relationship between two variables is a direct variation, inverse variation, or neither. How does the value of x change? increases decreases How does the value of y change? How does the value of y change? increases decreases decreases increases Test x and y values in direct variation model k = Test x and y values in inverse variation model: xy = k. Is each ratio of y to x equal to the same value, k? no yes Is each product of x and y equal to the same value, k? no yes neither direct variation inverse variation neither Problem Do the data in the table represent a direct variation, inverse variation, or neither? x 1 2 4 5 y 20 10 5 4 As the value of x increases, the value of y decreases, so test the table values in the inverse variation model: xy=k: 1.20 = 20, 2 - 10 = 20, 4.5=20, 5.4 = 20. Each product equals the same value, 20, so the data in the table model an inverse variation. Do the data in the table represent a direct variation, inverse variation, or neither? 1. 2. x 5 10 15 20 x 1 3 4 6 y 10 20 30 40 y 12 4 3 2 Pearson Texas Algebra II Inverse Variation To solve problems involving inverse variation, you need to solve for the constant of variation k before you can find an answer. Problem The time that is necessary to complete a task varies inversely as the number of people p working. If it takes 4 h for 12 people to paint the exterior of a house, how long does it take for 3 people to do the same job? k р Write an inverse variation. Because time is dependent on people, t is the dependent variable and p is the independent variable. 4 k 12 Substitute 4 fort and 12 for p. 48=k Multiply both sides by 12 to solve for k, the constant of variation. 4- 48 р Substitute 48 for k. This is the equation of the inverse variation. 48 = 16 3 Substitute 3 for p. Simplify to solve the equation. It takes 3 people 16 h to paint the exterior of the house. 3. The time t needed to complete a task varies inversely as the number of people p. It takes 5 h for seven men to install a new roof. How long does it take ten men to complete the job? 4. The time t needed to drive a certain distance varies inversely as the speed r. It takes 7.5 h at 40 mi/h to drive a certain distance. How long does it take to drive the same distance at 60 mi/h? 5. The cost of each item bought is inversely proportional to the number of items when spending a fixed amount. When 42 items are bought, each costs $1.46. Find the number of items when each costs $2.16. 6. The length 1 of a rectangle of a certain area varies inversely as the width w. The length of a rectangle is 9 cm when the width is 6 cm. Determine the length if the width is 8 cm.
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