Solve the problem.
-Mari drops a ball from a height of 11 meters and notices that on each bounce the ball returns to about $\frac { 8 } { 9 }$ of its previous height. About how far will ball travel before it comes to rest?
A) 187 m
B) 198 m
C) 99 m
D) 23.4 m

Question

Quizplus · Accepted Answer

The total distance the ball travels before coming to rest 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, the first term $a$ is the initial drop of 11 meters, and the common ratio $r$ is $\frac{8}{9}$, representing the fraction of the height the ball reaches on each bounce. However, since the ball travels both down and up on each bounce after the first, we must account for this by doubling the series after the first term. Thus, the total distance is $11 + 2(\frac{11 \cdot \frac{8}{9}}{1 - \frac{8}{9}}) = 11 + 2(\frac{11 \cdot \frac{8}{9}}{\frac{1}{9}}) = 11 + 2(11 \cdot 8) = 11 + 176 = 187$ meters.

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