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Q. 80

Expert-verified
Found in: Page 616

### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

# Whenever a certain ball is dropped, it always rebounds to a height p% (0 < p < 100) of its original position. What is the total distance the ball travels before coming to rest when it is dropped from a height of h meters?

The total distance the ball travels before coming to rest when it is dropped from a height of h meters is $h+\frac{ph}{50}\left(\frac{100}{100-p}\right).$

See the step by step solution

## Step 1. Given Information.

The ball after dropping rebounds to a height p% (0 < p < 100) of its original position.

## Step 2. Find the distance the ball travels before coming to rest when it is dropped from a height of h meters.

To find the total distance traveled by the ball, let's first find the distance traveled by the ball before coming to rest, each successive bounce has two times p% of the height as it involves up and down of the ball,

$d=h+2×h×\left(\frac{p}{100}\right)+2×h×{\left(\frac{p}{100}\right)}^{2}+...\phantom{\rule{0ex}{0ex}}d=h+2×h×\frac{p}{100}\left(1+\left(\frac{p}{100}\right)+{\left(\frac{p}{100}\right)}^{2}+...\right)\phantom{\rule{0ex}{0ex}}\mathrm{Use}\mathrm{the}\mathrm{formula}\mathrm{of}\mathrm{geometric}\mathrm{series}\left(\frac{\mathrm{a}}{1-\mathrm{r}}\right)\phantom{\rule{0ex}{0ex}}\mathrm{d}=h+2×h×\frac{p}{100}\left(\frac{1}{1-\frac{p}{100}}\right)\phantom{\rule{0ex}{0ex}}\mathrm{d}=h+\frac{ph}{50}\left(\frac{100}{100-p}\right)$

Thus, the total distance traveled by the ball before coming to the rest is $h+\frac{ph}{50}\left(\frac{100}{100-p}\right).$