6.4 Variation 325 Joint Variation One quantity may vary directly as a product of two or more other quantities. This type of variation is called joint variation . Example 6 Using the Constant of Proportionality For a certain chemistry application, V varies inversely as P. If V 2 = when P 26, = determine V when P 4. = Solution First, write the inverse variation equation. Then substitute V 2 = and P 26 = into the variation equation and solve for k. V k P k k 2 26 52 = = = Thus, the constant of proportionality, k, is 52. Now we substitute 52 for k and 4 for P into the original inverse variation equation to determine V. V k P V V 52 4 13 = = = Thus, when P is 4 V is 13. 7 Now try Exercise 45 Profile in Mathematics Danica McKellar Danica McKellar is an American actress, best known for her role as Winnie Cooper in the late 1980s television show The Wonder Years . After the show ended, McKellar took a break from acting and earned a Bachelor of Science degree in Mathematics from UCLA, where she co-authored a paper with a professor and a fellow student. The paper is nicknamed the Chayes-McKellarWinn theorem after the authors. In addition to her career in acting, McKellar has authored several books aimed at making mathematics less daunting for middle and high school girls. Her books include Math Doesn't Suck: How to Survive Middle School Math without Losing Your Mind or Breaking a Nail, Kiss My Math: Showing Pre-Algebra Who's Boss, Hot X: Algebra Exposed, and Girls Get Curves: Geometry Takes Shape . McKellar has also written several mathematics-based children’s books and maintains a website on which she features many entertaining aspects of mathematics. When asked about her dual role as an actress and mathematics advocate, she replied, “I recognize that I have a unique position to be a role model to young girls because I am doing something that they consider glamorous, which is acting, and yet I took the time to really get my education and study mathematics, and I think math is the cat's meow. And I share that I do.” Joint Variation If a variable y varies jointly with variables x and z, then y kxz = where k is the constant of proportionality. Example 7 Joint Variation in Geometry The area, A, of a triangle varies jointly as its base, b, and height, h. If the area of a triangle is 48 in.2 when its base is 12 in. and its height is 8 in., determine the area of a triangle whose base is 15 in. and whose height is 20 in. Solution First write the joint variation, then substitute the known values and solve for k. A kbh k k k k 48 (12)(8) 48 (96) 48 96 1 2 = = = = = Featureflash Photo Agency/Shutterstock
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