Difference between revisions of "2008 AMC 12B Problems/Problem 11"

(Solution)
(Solution: latexed answer)
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Plugging in our given condition, <math>1/8 = Height^3 \Rightarrow Height = 1/2</math>
 
Plugging in our given condition, <math>1/8 = Height^3 \Rightarrow Height = 1/2</math>
  
<math>8000*1/2=4,000</math>, answer choice A.
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<math>8000*1/2=4,000 \Rightarrow \boxed{A}</math>
  
 
==See Also==
 
==See Also==
 
{{AMC12 box|year=2008|ab=B|num-b=10|num-a=12}}
 
{{AMC12 box|year=2008|ab=B|num-b=10|num-a=12}}

Revision as of 12:30, 30 May 2011

Problem 11

A cone-shaped mountain has its base on the ocean floor and has a height of 8000 feet. The top $\frac{1}{8}$ of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain in feet?

$\textbf{(A)}\ 4000 \qquad \textbf{(B)}\ 2000(4-\sqrt{2}) \qquad \textbf{(C)}\ 6000 \qquad \textbf{(D)}\ 6400 \qquad \textbf{(E)}\ 7000$

Solution

In a cone, radius and height each vary inversely with increasing height (i.e. the radius of the cone formed by cutting off the mountain at $4,000$ feet is half that of the original mountain). Therefore, volume varies as the inverse cube of increasing height (expressed as a percentage of the total height of cone): $V_I*Height^3 = V_N$

Plugging in our given condition, $1/8 = Height^3 \Rightarrow Height = 1/2$

$8000*1/2=4,000 \Rightarrow \boxed{A}$

See Also

2008 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 10
Followed by
Problem 12
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All AMC 12 Problems and Solutions