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

Answer:

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

Step-by-step 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 $\frac{3}{4}$

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

$T_{n}=36\times (\frac{3}{4})^{n}$

If this term is less than 1

$36\times (\frac{3}{4})^{n}$

Taking log on both the sides

$log[36\times (\frac{3}{4})^{n}]$

$log36+nlog(\frac{3}{4} )$

1.5563 + n(-0.1249) < 0

0.1249n > 1.5563

n > $\frac{1.5563}{0.1249}$

n > 12.46

n ≈ 13

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

Answer 2

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