SOLUTION: An inverted right circular conical tank has an altitude equal to inches and a base with radius 1.25 inches. Find the dimensions and the volume of a similar tank whose volume is thr

Question 242496: An inverted right circular conical tank has an altitude equal to inches and a base with radius 1.25 inches. Find the dimensions and the volume of a similar tank whose volume is three times as much as the first conical tank.
I just got the volume of the conical tank which is (175pi)/12 cubic inches and its tripled volume is (175pi)/4 cubic inches. I'm not sure with the answer that I have arrived and if my answer is right, I don't know how to get the height and its radius. Please show your solution. Thanks a lot
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

In the first place, you didn't provide the actual value of the height. However, the problem can be solved in general terms if (and only if) the meaning of "similar" as used in your question is the technical meaning of "similar" as defined for similar solids.

The volume of a right circular cone, inverted or otherwise, is given by:

So, if we let

and

be the base radius and height of the smaller cone, and if we let

and

be the base radius and height of the larger cone, and furthermore if we let

and

be the ratio of the smaller radius to the larger radius and the smaller height to the larger height respectively.

Then, the smaller cone has a volume of:

and the larger cone has a volume of:

But since we have defined the ratios we can say that:

and

Substituting:

Since

, we can see that

However, since the radii and heights of the two cones must be in the same proportion in order that the cones be similar, we can say:

Substituting:

Cross-multiply:

Hence

Substituting:

If the dimensions of the original cone are

and

, then the dimensions of a similar cone with 3 times the volume are

and

But 1.44 should suffice for your purposes. Round any numerical approximation answer to two decimal places -- that is the least precision in your input measurements.

John

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