412 CHAPTER 3 Polynomial and Rational Functions Sometimes y varies as a power of x. If n is a positive integer greater than or equal to 2, then y is a greater-power polynomial function of x. For example, the area of a square of side x is given by the formula = x2, so the area varies directly as the square of the length of a side. Here k = 1. EXAMPLE 1 Solving a Direct Variation Problem The area of a rectangle varies directly as its length. If the area is 50 m2 when the length is 10 m, find the area when the length is 25 m. (See Figure 68.) SOLUTION Step 1 The area varies directly as the length, so = kL, where represents the area of the rectangle, L is the length, and k is a nonzero constant. Step 2 Because = 50 when L = 10, we can solve the equation = kL for k. 50 = 10k Substitute for and L. k = 5 Divide by 10. Interchange sides. Step 3 Using this value of k, we can express the relationship between the area and the length as follows. = 5L Direct variation equation Step 4 To find the area when the length is 25, we replace L with 25. = 5L = 51252 Substitute for L. = 125 Multiply. The area of the rectangle is 125 m2 when the length is 25 m. S Now Try Exercise 27. Solving a Variation Problem Step 1 Write the general relationship among the variables as an equation. Use the constant k. Step 2 Substitute given values of the variables and find the value of k. Step 3 Substitute this value of k into the equation from Step 1, obtaining a specific formula. Step 4 Substitute the remaining values and solve for the required unknown. Direct Variation as n th Power Let n be a positive real number. Then y varies directly as the nth power of x, or y is directly proportional to the nth power of x, if for all x there exists a nonzero real number k such that y =kxn. ! = ? 25 m ! = 50 m2 10 m Figure 68
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