Question

A rubber ball is dropped from a height of 29 feet, and on each bounce it rebounds up 84% of its previous height. Assuming the ball bounces indefinitely, find the total vertical distance traveled. Round your answer to two decimal places.

I think it's 333.5 but I'm not sure.

Answer 1

Answer:

333.5 feet

Step-by-step explanation:

In the first trip to the ground, the ball goes 29 feet. After that, it bounces until it reaches 0.84 times 29 feet and goes back to ground the same distance. It makes the total distance:

$29+58*0.84$

For the second bounce, the next height will be 0.84 of the previous one, that is $58*0.84*0.84=58*0.84^2$ and the total distance:

$29+58*0.84+58*0.84^2$

The game continues such that for n bounds the total distance will be

$29+\sum 58*0.84^n=29+58\sum 0.84^n$

When n goes to infinity, the sum becomes a sum of an infinite series

$\sum_{n=0}^{\infty }ar^n=\frac{a}{1-r}$

In our case, a=0.84 and r=0.84

The sum of all bounces will be

$S=29+58\sum 0.84^n=29+\frac{0.84}{1-0.84}=333.5\ feet$