SOLUTION: a super-bouncy ball rebounds half the height it drops. the ball is dropped from a height of 176 feet. How far off the ground is the ball when it has travled a total of 500 feet?

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Question 1073879: a super-bouncy ball rebounds half the height it drops. the ball is dropped from a height of 176 feet. How far off the ground is the ball when it has travled a total of 500 feet?

Answer by rothauserc(4718) (Show Source):
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The falls form a geometric series with initial term of 176 and a common ratio of (1/2)
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The general form for the sum of a geometric series is
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S(n) = a1 * (1-r^n) / (1-r)
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For the falls, a(1) = 176 and r = (1/2)
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S(n) = 176 * (1 - (1/2)^n) / (1 - (1/2))
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For the rebounds, a(1) = 88 and r = (1/2)
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S(n) = 88 * (1 - (1/2)^n) / (1 - (1/2))
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We know
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500 = (352 * (1 - (1/2)^(n+1))) + (176 * (1 - (1/2)^n))
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divide both sides of = by 4
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125 = (88 * (1 - (1/2)^(n+1))) + (44 * (1 - (1/2)^n))
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125 = (88 * (1 - 2^(-n - 1))) + (44 * (1 - 2^(-n)))
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7 * 2^n = 132
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use definition of logarithm
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n = 4.237
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We calculate the rebounds for n = 4 and n = 3
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S(4) = 88 * (1 - (1/2)^4) / (1 - (1/2)) = 165
S(3) = 88 * (1 - (1/2)^(4.237)) / (1 - (1/2)) = 154
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The ball is 165 - 154 = 11 feet above the ground
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