324 CHAPTER 6 Algebra, Graphs, and Functions At 100 miles per hour, it would take 1 hour to cover this 100 mile distance. At a rate of 50 miles an hour, it would take 2 hours. At a rate of 25 miles an hour, it would take 4 hours. Note that as the rate (or speed) decreases, the time increases and vice versa. The preceding equation can be written t r 100 = This equation is an example of an inverse variation equation. The time and rate are inversely proportional. The constant of proportionality in this case is 100. Inverse Variation If a variable y varies inversely with a variable x, then y k x = where k is the constant of proportionality. Two quantities vary inversely, or are inversely proportional, when as one quantity increases the other quantity decreases and vice versa. Examples 5 and 6 illustrate inverse variation. Example 5 Inverse Variation in Speaker Loudness The loudness, L, of a stereo speaker, measured in decibels (dB), varies inversely as the square of the distance, d, of the listener from the speaker. a) If L 20 dB = when d 6 feet, = determine the constant of proportionality. b) Determine the loudness 4 feet from the speaker. Solution a) Since L varies inversely as the square of d, we start with the equation L . k d2 = Next, to determine k, substitute 20 for L and 6 for d into the variation equation and then solve for k. L k d k k k k 20 6 20 36 (20)(36) 720 2 2 = = = = = Thus, the constant of proportionality, k, is 720. b) To determine the loudness at 4 feet, we start with the variation equation, L , k d2 = and substitute k 720 = and d 4. = L k d 720 4 720 16 45 2 2 = = = = Thus, the loudness at 4 feet is 45 decibels. 7 Now try Exercise 43 26kot/Shutterstock
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