A rubber ball is dropped from a height of 60 feet. If it rebounds approximately two-thirds the distance after each fall, use an infinite geometric series to approximate the total distance the ball travels.
A)300 ft
B)480 ft
C)240 ft
D)60 ft
E)180 ft

Question

Quizplus · Accepted Answer

The total distance traveled by the ball can be found using the formula for the sum of an infinite geometric series, $S = \frac{a}{1 - r}$, where $a$ is the first term and $r$ is the common ratio. Here, $a = 60$ feet (the initial drop) and $r = \frac{2}{3}$ (since it rebounds two-thirds the distance each time). The series for the total distance includes both the downward and upward movements after the initial drop, so the first upward movement is $60 \times \frac{2}{3} = 40$ feet, making the series 60, 40, 40(2/3), 40(2/3)^2, ..., for both the downward and upward distances after the first drop. The effective first term for the series after the initial drop is $40$ (since the initial 60 feet is only traveled once), and the series becomes 60 + 2(40 + 40(2/3) + 40(2/3)^2 + ...). The sum of the infinite series after the initial drop is $S = \frac{40}{1 - \frac{2}{3}} = 120$ feet. Therefore, the total distance is $60 + 2 \times 120 = 300$ feet.

A Rubber Ball Is Dropped from a Height of 60