322 CHAPTER 6 Algebra, Graphs, and Functions The formula used to calculate distance traveled is Distance rate time = ⋅ Since the rate is a constant 40 miles per hour, the formula can be written d t 40 = We say that distance varies directly as time or that distance is directly proportional to time. The preceding equation is an example of a direct variation equation. Direct Variation If a variable y varies directly with a variable x, then y kx = where k is the constant of proportionality (or the variation constant). In our discussion above, d varies directly as t, and 40 is the constant of proportionality. Examples 1 through 4 illustrate direct variation. Example 1 Direct Variation in Physics The length that a spring will stretch, S, varies directly with the force (or weight), F, attached to the spring. Write the equation for the length that a spring will stretch, S, if the constant of proportionality is 0.07. Solution S kF = S varies directly as F. S F 0.07 = Constant of proportionality, k, is 0.07. 7 Now try Exercise 19 Example 2 Direct Variation in Medicine The recommended dosage, d, of the antibiotic drug vancomycin is directly proportional to a person’s weight, w. a) Write this variation equation. b) Determine the recommended dosage, in milligrams, for Jayce, who weighs 128 lb. Assume the constant of proportionality for the dosage is 18. Solution a) d kw = b) d 18(128) 2304 = = The recommended dosage for Jayce is 2304 mg. 7 Now try Exercise 23 In certain variation problems, the constant of proportionality, k, may not be known. In such cases, we can often determine it by substituting the given values in the variation formula and solving for k. Example 3 Falling Distance of a Dropped Object The distance an object falls when dropped from a building, d, varies directly as the square of the time, t, after the object is dropped. a) If d 144 = feet when t 3 = seconds, determine the constant of proportionality. b) Determine the distance the object falls after 2 seconds. Eggeeggjiew/123RF
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