Question

A rubber ball is dropped from a 100 ft tall building. Each time it bounces, it rises to threequarters its previous height. So, after its first bounce it rises to 75 ft, and after its second
bounce it rises to 3/4 of 75 ft, and so on forever. What is the total distance the ball travels?

Answer 1

geometry series

\[\sum_{n=0}^{\infty}75(.75)^n\]

Answer 2

use formula

\[\frac{a_0}{1-r}\]

Answer 3

\[ 100 + 2 \times 75 \sum_{n=0}^\infty (3/4)^n \]
First it falls 100, then it travels 75 twice, then 75(3/4) twice, then 75(3/4)^2 twice, etc.

Answer 4

Make sense? If so, what final answer do you get?

Answer 5

and \( \sum \limits_{n=0}^\infty \left(\frac 3 4 \right)^n = \large \frac{ \frac3 4 }{1 - \frac3 4 }\) which is sum of infinite series, this is convergent as common ratio is less than 1.

Answer 6

sum of infinite *geometric* series

Answer 7

ffm, careful with the numerator.

Answer 8

75, not 3/4

Answer 9

oops, first term is 1; \[ \sum \limits_{n=0}^\infty \left(\frac 3 4 \right)^n = \large \frac{ 1 }{1 - \frac3 4 } \]

Answer 10

so the answer is 700 ft.

Answer 11

Thanks. Yes I understand how it worked out. I did \[100 + \frac{2 \times 75}{1 - \frac{3}{4}}\]