Question

A ball is dropped from a height of 36 feet. At each bounce the ball reaches a height that is three quarters of the previous height. How many bounces must the ball make before it rebounds less than 1 foot?

Answer 1

Before it rebounds n times we have the inequality

36 (3/4)^n < 1

(3/4)^n < 1/36

n ln 3/4 < ln 1/36

n < ln 1/36 / ln 3/4

n < 12.46

so n = 12

answer is 12 bounces

Answer 2

Ball will make 13 bounces before it rebounds less than 1 foot.

Explanation:

A ball is dropped from a height of 36 feet.

In each bounce the ball reaches a height that is three quarters of the previous height.

Sequence formed will be 36, 27, 20.25.........

This sequence has a common ratio of
`(3)/(4)`

Therefore, sequence will be a geometric sequence.

Explicit formula of this sequence will be

`T_(n)=a(r)^(n)`

where a = first term of the sequence

r = common ratio

n = number of term

For this sequence formula will be

`T_(n)=36* ((3)/(4))^(n)`

If this term is less than 1

`36* ((3)/(4))^(n)<1`

Taking log on both the sides

`log[36* ((3)/(4))^(n)]<log1`

`log36+nlog((3)/(4) )<log1`

1.5563 + n(-0.1249) < 0

0.1249n > 1.5563

n >
`(1.5563)/(0.1249)`

n > 12.46

n ≈ 13

Therefore, ball will make 13 bounces before it rebounds less than 1 foot.