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