2008 AMC 12B Problems/Problem 11: Difference between revisions
New page: ==Problem 11== A cone-shaped mountain has its base on the ocean floor and has a height of 8000 feet. The top <math>\frac{1}{8}</math> of the volume of the mountain is above water. What is ... |
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==Solution== | ==Solution== | ||
In a cone, radius and height each vary inversely. with increasing height (the radius at <math>4,000</math> feet is half that at <math>0</math> feet). Therefore, volume varies as the inverse cube of increasing height (expressed as a percentage of the total height of cone): | In a cone, radius and height each vary inversely. with increasing height (the radius at <math>4,000</math> feet is half that at <math>0</math> feet). Therefore, volume varies as the inverse cube of increasing height (expressed as a percentage of the total height of cone): | ||
<math> | <math>V_I*Height^3 = V_N</math> | ||
Plugging in our given condition, <math>1/8 = Height^3 | 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 | |||
<math>8000*1/2=4,000</math>, answer choice A. | |||
Revision as of 22:11, 1 March 2008
Problem 11
A cone-shaped mountain has its base on the ocean floor and has a height of 8000 feet. The top 
of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain in feet?
Solution
In a cone, radius and height each vary inversely. with increasing height (the radius at 
feet is half that at 
feet). Therefore, volume varies as the inverse cube of increasing height (expressed as a percentage of the total height of cone):
Plugging in our given condition, 
, answer choice A.
