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A right cylinder and a cube have the same surface area...

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A right cylinder and a cube have the same surface area...

by AAPL » Wed Nov 29, 2017 7:04 pm
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%

The OA is B.

Please, can any expert assist me with this PS question? I don't have it clear and I appreciate if any explain it for me. Thanks.
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by EconomistGMATTutor » Sat Dec 02, 2017 6:47 am
Hello AAPL.

Let's see.

The surface area of the cylinder is: $$A_{cy}=2\pi\cdot r^2+2\pi\cdot r\cdot h$$ and the surface area of the cube is: $$A_{cu}=6L^2.$$ As the height of the cylinder is equal to its diameter, then $$h=2r,\ so\ A_{cy}=2\pi r^2+2\pi\cdot r\cdot2r=2\pi r^2+4\pi r^2=6\pi r^2.$$ Now, as the cylinder and the cube have the same surface area then $$6\pi r^2=6L^2\leftrightarrow\ \pi r^2=L^2\ \leftrightarrow\ L=r\sqrt{\pi}.$$ Now, the volumen of the cylinder and the cube are: $$V_{cy}=\pi r^2\cdot h=2\pi r^3,\ \ \ \ \ \ V_{cu}=L^3.$$ Replacing L in this last equation we get: $$V_{cu}=L^3=\left(r\sqrt{\pi}\right)^3=\pi r^3\sqrt{\pi}=\frac{2\pi r^3\sqrt{\pi}}{2}=\frac{V_{cy}\sqrt{\pi}}{2}\approx V_{cy}\cdot0.88=88\%V_{cy}.$$ That implies that the volumen of the cylinder is approximately 12% greater than the volume of the cube.

Whatching the options we have that the correct answer is B.

I hope this explanation may help you.

Feel free to ask me again if you have any doubt.
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by Scott@TargetTestPrep » Sun Sep 29, 2019 6:21 pm
AAPL 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 = 3Ï€d^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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