SOLUTION: A paper cup is in the shape of a right circular cone with a diameter of 8cm and a height of 12cm. a.) Suppose the cup is filled with water to a depth of 8cm. Calculate the volum

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Question 1143219: A paper cup is in the shape of a right circular cone with a diameter of 8cm and a height of 12cm.
a.) Suppose the cup is filled with water to a depth of 8cm. Calculate the volume of water in the cup. Round to the nearest whole number. (HINT: Use similar triangles.)
b.) To what depth must the cup be filled in order to be half full of water? Round to the nearest tenth.
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The volume of the cup is



a. If the cup is filled to a depth of 8cm, that is 2/3 the full depth, so the radius of the water in the cup will be 2/3 of the full radius, which is 8/3cm. The volume of water is then

V+=+%281%2F3%29%28pi%29%28%288%2F3%29%5E2%29%288%29+=+%28512%2F27%29pi

Since the question asks for an answer as the nearest whole number, perform the calculation and round as required.

Note that, though the problem gives a hint about using similar triangles, there is a much faster way to answer this question. If the height of the water in the cup is 2/3 the full height of the cup, then the volume of water in the cup is (2/3)^3 of the total volume:

%28%282%2F3%29%5E3%29%2864pi%29+=+%28512%2F27%29pi

b. This question is answered far more easily using the concept noted above.

If the volume is 1/2 of the total volume, then the height of the cone (depth of the water) is the full height, multiplied by the CUBE ROOT of 1/2.

h+=+12%2A%281%2F2%29%5E%281%2F3%29+=+9.5244 to 4 decimal places.

Than round as directed.