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After falling, a rubber ball bounces back to one-quarter of its previous height.
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Given that the ball fell from a height of 653 centimeters above the ground, find to the nearest integer the height it would reach after its second bounce.
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We’ve been given information about a rubber ball being dropped from a height of 653 centimeters.
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We’re told that it bounces back to just one-quarter of its previous height each time and asked to find the height it would reach after its second bounce.
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Now, we’re going to need to be a little bit careful here.
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If we take the starting height as 653 centimeters, its second height is immediately after the first bounce, and its third height is immediately after the second bounce.
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So, this is what we’re interested in finding.
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And in fact, if we look carefully, we see we’re looking at a geometric sequence or a geometric progression.
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The terms in a geometric progression are found by multiplying the previous one by a fixed nonzero number, which we call the common ratio.
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The 𝑛th term of a geometric progression is found by 𝑎 times 𝑟 to the power of 𝑛 minus one.
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𝑎 is the first term of the sequence and 𝑟 is its common ratio.
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So, let’s define 𝑎 and 𝑟 for our sequence.
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We can say that the first height 𝑎 is equal to 653 or 653 centimeters.
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The common ratio is one-quarter.
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It bounces back to just one-quarter of the height from the previous bounce.
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And in fact, we said we’re going to let 𝑛 be equal to three.
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The height we’re interested in immediately after the second bounce is the third height.
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The third height is, therefore, 𝑎 sub three.
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And it’s given by 653 — remember, that’s the first term — multiplied by a quarter — that’s the common ratio — to the power of 𝑛 minus one, which is three minus one or 653 times a quarter squared.
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To find the square of one-quarter, we simply square the numerator and separately square the denominator.
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So, the third height is 653 times one sixteenth, which is 40.8125 or 40.8125 centimeters.
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Remember, we’re told to give our answer to the nearest integer, the nearest whole number.
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And 40.8125 rounded to the nearest whole number is 41.
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So, the height it reaches after its second bounce is 41 centimeters.