Bunuel wrote:
A right cylinder and a cube have the same surface area. If the height of the cylinder is equal to its diameter, then the volume of the cylinder is approximately what percent greater than the volume of the cube?
A. 6%
B. 13%
C. 26%
D. 32%
E. 35%
We can let d = the diameter (or height) of the cylinder. Since the surface area of a cylinder is 2πrh + 2πr^2, the surface area of this cylinder is 2π(d/2)(d) + 2π(d/2)^2 = πd^2 + πd^2/2 = 3πd^2/2.
Since the cube has the same surface area and the surface area of a cube is 6s^2, we have:
6s^2 = 3πd^2/2
s^2 = πd^2/4
s = (√π)d/2
Since the volume of a cylinder is πr^2h, the volume of this cylinder is π(d/2)^2*d = πd^3/4. Since the volume of a cube is s^3, the volume of this cube is [(√π)d/2]^3 = (π√π)d^3/8. Therefore, the volume of the cylinder is:
(πd^3/4)/[(π√π)d^3/8] = (1/4)/(√π/8) = 2/√π ≈ 1.13 times the volume of the cube.
In other words, the volume of the cylinder is approximately 13% greater than the volume of the cube.
Answer: B
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