A solid cube has a total surface area of S and a volume of V. A solid right circular cylin… — GRE General Test

A solid cube has a total surface area of $S$ and a volume of $V$ . A solid right circular cylinder has a height equal to its base diameter. If the total surface area of the cylinder is also equal to $S$ , what is the volume of the cylinder in terms of $V$ ?

$\frac{2}{\sqrt{\pi}} V$ Answer
B
$\frac{\sqrt{\pi}}{2} V$
C
$\frac{1}{\sqrt{\pi}} V$
D
$\sqrt{\frac{27}{2\pi}} V$
E
$\frac{2}{\pi} V$

Answer

\frac{2}{\sqrt{\pi}} V

The total surface area of a cube with edge

a

6a^2

and its volume is

V = a^3

. For a cylinder with base radius

r

and height

h = 2r

, the total surface area is

2\pi r^2 + 2\pi r(2r) = 6\pi r^2

. Setting

6\pi r^2 = 6a^2

yields

r = \frac{a}{\sqrt{\pi}}

and

h = \frac{2a}{\sqrt{\pi}}

. Substituting these into the volume formula

V_{\text{cyl}} = \pi r^2 h

produces

\pi \left(\frac{a^2}{\pi}\right) \left(\frac{2a}{\sqrt{\pi}}\right) = \frac{2a^3}{\sqrt{\pi}} = \frac{2}{\sqrt{\pi}} V

Step-by-Step Solution

Express the surface area and volume of the cube in terms of its side length

a

Surface area

S = 6a^2

and volume

V = a^3

A cube with edge length

a

has

6

identical square faces of area

a^2

and volume

a^3

Set up the total surface area formula for the cylinder with radius

r

and height

h = 2r

, and equate it to

S

Total surface area

S_{\text{cyl}} = 2\pi r^2 + 2\pi r h = 2\pi r^2 + 2\pi r (2r) = 6\pi r^2 = 6a^2

The cylinder's height is equal to its base diameter (

2r

). Equating surface areas gives

6\pi r^2 = 6a^2

Solve for radius

r

in terms of edge length

a

r^2 = \frac{a^2}{\pi} \implies r = \frac{a}{\sqrt{\pi}}

Dividing both sides by

6\pi

and taking the square root isolates

r

Calculate the volume of the cylinder in terms of

V

V_{\text{cyl}} = \pi r^2 h = \pi \left(\frac{a^2}{\pi}\right) \left(\frac{2a}{\sqrt{\pi}}\right) = \frac{2a^3}{\sqrt{\pi}} = \frac{2}{\sqrt{\pi}} V

Substituting

r^2 = \frac{a^2}{\pi}

and

h = \frac{2a}{\sqrt{\pi}}

into the cylinder volume formula

\pi r^2 h

yields the answer in terms of

V = a^3

Key Concept

Volume and surface area relationship between geometric solids

Estimated Time:2m 0s

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