Difference between revisions of "2008 AMC 8 Problems/Problem 12"

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\textbf{(D)}\ 6 \qquad
 
\textbf{(D)}\ 6 \qquad
 
\textbf{(E)}\ 7</math>
 
\textbf{(E)}\ 7</math>
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==Solution==
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Each bounce is <math>2/3</math> times the height of the previous bounce. The first bounce reaches <math>2</math> meters, the second <math>4/3</math>, the third <math>8/9</math>, the fourth <math>16/27</math>, and the fifth <math>32/81</math>. Half of <math>81</math> is <math>40.5</math>, so the ball does not reach the required height on bounce <math>\boxed{\textbf{(C)}\ 5}</math>.
  
 
==See Also==
 
==See Also==
 
{{AMC8 box|year=2008|num-b=11|num-a=13}}
 
{{AMC8 box|year=2008|num-b=11|num-a=13}}
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{{MAA Notice}}

Revision as of 01:30, 5 July 2013

Problem

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?

$\textbf{(A)}\  3 \qquad \textbf{(B)}\  4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$

Solution

Each bounce is $2/3$ times the height of the previous bounce. The first bounce reaches $2$ meters, the second $4/3$, the third $8/9$, the fourth $16/27$, and the fifth $32/81$. Half of $81$ is $40.5$, so the ball does not reach the required height on bounce $\boxed{\textbf{(C)}\ 5}$.

See Also

2008 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 11
Followed by
Problem 13
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AJHSME/AMC 8 Problems and Solutions

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