A ball drops from a height of 6 ft. With each bounce, the ball rebounds to $\frac { 2 } { 5 }$ of its height. The total vertical distanced traveled is given by $6 + 6 \left( \frac { 2 } { 5 } \right) + 2 ( 6 ) \left( \frac { 2 } { 5 } \right) ^ { 2 } + \ldots$ After the first term, the series is an infinite geometric series. Compute the total vertical distance traveled.
A) The total vertical distance traveled is 10 ft.
B) The total vertical distance traveled is 14 ft.
C) The total vertical distance traveled is 4 ft.
D) The total vertical distance traveled is 8 ft.

Question

Quizplus · Accepted Answer

The total distance is the sum of the initial drop (6 ft) and the infinite geometric series for the bounces. The series has a first term of $6 \times \frac{2}{5}$ and a common ratio of $\left(\frac{2}{5}\right)^2$. The sum of the series is $\frac{6 \times \frac{2}{5}}{1 - \left(\frac{2}{5}\right)^2}$. Adding the initial drop gives a total distance of 14 ft.

A Ball Drops from a Height of 6 Ft 25\frac { 2 } { 5 }52​

A Ball Drops from a Height of 6 Ft $\frac { 2 } { 5 }$