2008 AMC 8 Problem 12

Question

A ball is dropped from a height of

3

meters. On its first bounce it rises to a height of

2

meters. It keeps falling and bouncing to

\frac{2}{3}

of the height it reached in the previous bounce. On which bounce will it not rise to a height of

0.5

meters?

\ 3

\ 4

\ 5

\ 6

\ 7

List the bounce heights by multiplying by

\frac{2}{3}

each time.

Compare the fourth and fifth bounce heights with

\frac{1}{2}

.

The first bounce rises to

2

meters. Each later bounce is

\dfrac{2}{3}

of the previous bounce.

The fourth bounce rises to

2\left(\dfrac{2}{3}\right)^3=\dfrac{16}{27}

, which is greater than

\dfrac{1}{2}

.

The fifth bounce rises to

2\left(\dfrac{2}{3}\right)^4=\dfrac{32}{81}

, which is less than

\dfrac{1}{2}

.

Thus, C is the correct answer.

Problem 12 in other years