SOLUTION: Sand is piled into a right circular cone with a diameter of 12 feet and height of 8 feet. An additional 200 cubic feet of sand is added uniformly so that the cone of sand has the s

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Question 1095236: Sand is piled into a right circular cone with a diameter of 12 feet and height of 8 feet. An additional 200 cubic feet of sand is added uniformly so that the cone of sand has the same ratio of height to diameter as before. What is the new height of the sand pile?
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!

Think of the original pile of sand and the final pile as similar cones.

By a powerful general principle regarding similar figures, the ratio of the volumes of the two cones will be the cube of the scale factor between the two cones (ratio of measures of length).

So the ratio of the heights of the two cones (a measurement of length) will be the cube root of the ratio of their volumes.

Volume of original pile:
V1+=+%281%2F3%29%28pi%29%286%5E2%29%288%29
Volume of final pile:
V2+=+%281%2F3%29%28pi%29%286%5E2%29%288%29+%2B+200
Ratio of volumes:
V2%2FV1
Ratio of heights:
%28V2%2FV1%29%5E%281%2F3%29
Height of the final pile:
8%2A%28V2%2FV1%29%5E%281%2F3%29

By my calculations, the new height is 13.66 feet.