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A certain rectangular crate measures 8 feet by 10 feet by 12 feet. A cylindrical gas tank is to be made for shipment in the crate and will stand upright when the crate is placed on one of its six faces. What should the radius of the tank be if it is to be of the largest possible volume?

  1. 4
  2. 5
  3. 6
  4. 8
  5. 10

The answer is given = 5.

But what is the problem with radius 6. If we guess 6 is the radius then, the rectangle of the edge 8 or 10 could fit. I think, i have misunderstand the question.

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  • $\begingroup$ Cylinder needs to fit in the rectangle, not the other way around. $\endgroup$ – Karolis Juodelė Jul 30 '14 at 8:02
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Obviously cylinder has to fit in crate. For best usage of crate's space, diameter of cylinder should equal the smaller side of the crate's face. So three possibilities arise

1) with 8x10 base and 12 height gives volume pi*4*4*12 = pi*196;

2) with 8x12 base and 10 height gives volume pi*4*4*10 = pi*160;

3) with 12x10 base and 8 height gives volume pi*5*5*8 = pi*200;

thus with crate's base as 12x10 and height 8 we get maximum volume of cylinder, and the radius is half of the smaller side of base i.e 10/2 = 5.

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