A solid right circular cylinder has a base radius of r and a height of 4r. A solid sphere … — GRE General Test

A solid right circular cylinder has a base radius of $r$ and a height of $4r$ . A solid sphere has a radius of $R$ . If the total surface area of the sphere is equal to the total surface area of the cylinder, what is the ratio of the volume of the sphere to the volume of the cylinder?

$\frac{5\sqrt{10}}{12}$ Cevap
B
$\frac{\sqrt{10}}{2}$
C
$\frac{2\sqrt{2}}{3}$
D
$\frac{5\sqrt{10}}{6}$
E
$\frac{5}{3}$

Cevap

The ratio of the volume of the sphere to the volume of the cylinder is

\frac{5\sqrt{10}}{12}

The total surface area of the cylinder is the sum of its lateral area and two circular bases:

2\pi r(4r) + 2\pi r^2 = 10\pi r^2

. Setting this equal to the sphere's surface area

4\pi R^2

yields

R/r = \sqrt{5/2} = \sqrt{10}/2

. The ratio of the sphere's volume

\frac{4}{3}\pi R^3

to the cylinder's volume

\pi r^2 (4r) = 4\pi r^3

\frac{1}{3}(R/r)^3 = \frac{1}{3} \left(\frac{\sqrt{10}}{2}\right)^3 = \frac{5\sqrt{10}}{12}

Adım Adım Çözüm

Calculate the total surface area of the cylinder.

\text{TSA}_{\text{cyl}} = 2\pi r^2 + 2\pi r h = 2\pi r^2 + 2\pi r(4r) = 10\pi r^2

A cylinder's total surface area consists of two circular bases (

2\times\pi r^2

) and the lateral surface area (

2\pi r h

Equate the total surface area of the sphere to the total surface area of the cylinder to find the ratio of

R

r

4\pi R^2 = 10\pi r^2 \implies R^2 = \frac{5}{2}r^2 \implies \frac{R}{r} = \sqrt{\frac{5}{2}} = \frac{\sqrt{10}}{2}

The total surface area of a sphere of radius

R

4\pi R^2

Express the volumes of both figures in terms of

r

and

R

V_{\text{sph}} = \frac{4}{3}\pi R^3

and

V_{\text{cyl}} = \pi r^2 h = \pi r^2 (4r) = 4\pi r^3

The volume of a sphere is

\frac{4}{3}\pi R^3

and the volume of a cylinder is

\pi r^2 h

Compute the ratio of the volume of the sphere to the volume of the cylinder.

\frac{V_{\text{sph}}}{V_{\text{cyl}}} = \frac{\frac{4}{3}\pi R^3}{4\pi r^3} = \frac{1}{3}\left(\frac{R}{r}\right)^3 = \frac{1}{3}\left(\frac{\sqrt{10}}{2}\right)^3 = \frac{1}{3} \cdot \frac{10\sqrt{10}}{8} = \frac{5\sqrt{10}}{12}

Substitute

\frac{R}{r} = \frac{\sqrt{10}}{2}

into the ratio expression.

Anahtar Kavram

Relating 3D surface area formulas to volume formulas for cylinders and spheres

Tahmini Süre:2m 0s

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