Choose the one alternative that best completes the statement or answers the question.
-A ping-pong ball is dropped from a height of 9 ft and always rebounds 13 of the distance fallen. Find the total sum of the rebound heights of the ball.
A) 3 ft
B) 6 ft
C) 4.5 ft
D) 13.5 ft

Question

Quizplus · Accepted Answer

The total sum of the rebound heights forms a geometric series. The first term is the height of the first rebound, which is $ \frac{2}{3} \times 9 = 6 $ ft. The common ratio is $ \frac{2}{3} $, because the ball rebounds to $ \frac{2}{3} $ of the height it fell from each time. The sum of an infinite geometric series is given by $ S = \frac{a}{1 - r} $, where $ a $ is the first term and $ r $ is the common ratio. Substituting the values, we get $ S = \frac{6}{1 - \frac{2}{3}} = \frac{6}{\frac{1}{3}} = 18 $ ft. However, this calculation includes the initial drop of 9 ft, so we subtract that to get the total sum of the rebound heights: $ 18 - 9 = 9 $ ft. This seems to be a miscalculation in the explanation process. The correct approach to find the sum of the rebound heights (excluding the initial drop) should indeed consider the geometric series starting from the first rebound height of 6 ft, but the total distance covered by the ball includes both the ascent and descent of each rebound, effectively doubling the sum of the rebounds. The correct calculation for the sum of the rebounds alone (not including the initial drop) should directly apply the formula for the sum of an infinite geometric series without subtracting the initial drop, as the question specifically asks for the sum of the rebound heights, not the total distance traveled by the ball including the initial drop. Therefore, the correct explanation is that the sum of the rebound heights is a geometric series with the first term being 6 ft and the common ratio being $ \frac{2}{3} $, leading to a sum of rebounds that is part of the total distance the ball travels but calculated correctly with the geometric series formula. My initial explanation mistakenly subtracted the initial drop in an attempt to correct for the total distance, which was not asked for in the question. The correct answer should reflect the sum of the rebounds as calculated by the geometric series formula without subtracting the initial drop height, focusing solely on the rebounds.

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