413 3.7 Variation Inverse Variation Another type of variation is inverse variation. With inverse variation, where k 70, as the value of one variable increases, the value of the other decreases. This relationship can be expressed as a rational function. EXAMPLE 2 Solving an Inverse Variation Problem In a certain manufacturing process, the cost of producing a single item varies inversely as the square of the number of items produced. If 100 items are produced, each costs $2. Find the cost per item if 400 items are produced. SOLUTION Step 1 Let x represent the number of items produced and y represent the cost per item. Then, for some nonzero constant k, write the variation equation. y = k x2 y varies inversely as the square of x. Step 2 2 = k 1002 Substitute; y = 2 when x = 100. k = 20,000 Solve for k. Step 3 The relationship between x and y is y = 20,000 x2 . Step 4 When 400 items are produced, the cost per item is found as follows. y = 20,000 x2 = 20,000 4002 = 0.125 The cost per item is $0.125, or 12.5 cents. S Now Try Exercise 37. Combined and Joint Variation In combined variation, one variable depends on more than one other variable. Specifically, when a variable depends on the product of two or more other variables, it is referred to as joint variation. Inverse Variation as n th Power Let n be a positive real number. Then y varies inversely as the nth power of x, or y is inversely proportional to the nth power of x, if for all x there exists a nonzero real number k such that y = k xn . If n = 1, then y = k x , and y varies inversely as x. Joint Variation Let m and n be real numbers. Then y varies jointly as the nth power of x and the mth power of z if for all x and z there exists a nonzero real number k such that y =kxnzm.
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