A solid right circular cone has a base radius of 6\text{ centimeters} and a height of 8\te… — SAT

A solid right circular cone has a base radius of $6\text{ centimeters}$ and a height of $8\text{ centimeters}$ . A sphere is inscribed inside the cone such that it is tangent to the base of the cone and tangent to the lateral surface of the cone. What is the ratio of the volume of the sphere to the volume of the cone?

A
$\frac{1}{9}$
$\frac{3}{8}$ Answer
C
$\frac{1}{8}$
D
$\frac{8}{9}$

Answer

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

\frac{3}{8}

The correct answer is found by first determining the radius of the inscribed sphere. By analyzing the 2D cross-section of the cone, which is an isosceles triangle with a base of

12

and a height of

8

, the slant height is found to be

10

using the Pythagorean theorem. The radius of the inscribed circle (the sphere's radius) is calculated using the area and semiperimeter of the triangle, yielding

R = 3

. The volume of the sphere is

36\pi

and the volume of the cone is

96\pi

. Dividing the sphere's volume by the cone's volume gives the ratio of

\frac{3}{8}

Step-by-Step Solution

Find the slant height of the cone.

Slant height

L = 10\text{ centimeters}

The vertical cross-section of the cone is an isosceles triangle with base

2r = 12

and height

h = 8

. The slant height is the hypotenuse of the right triangle formed by the radius, height, and slant height:

L = \sqrt{6^2 + 8^2} = 10

Calculate the radius of the inscribed sphere.

Sphere radius

R = 3\text{ centimeters}

The sphere's radius is the inradius of the cross-sectional isosceles triangle. The area of this triangle is

A = \frac{1}{2} \times 12 \times 8 = 48

, and its semiperimeter is

s = \frac{12 + 10 + 10}{2} = 16

. Using the inradius formula

R = \frac{A}{s}

, we get

R = \frac{48}{16} = 3

Calculate the volume of the cone.

Volume of the cone

V_{\text{cone}} = 96\pi\text{ cubic centimeters}

Using the volume of a cone formula,

V_{\text{cone}} = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi (6^2)(8) = 96\pi

Calculate the volume of the sphere.

Volume of the sphere

V_{\text{sphere}} = 36\pi\text{ cubic centimeters}

Using the volume of a sphere formula,

V_{\text{sphere}} = \frac{4}{3}\pi R^3 = \frac{4}{3}\pi (3^3) = 36\pi

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

Ratio

= \frac{3}{8}

Divide the volume of the sphere by the volume of the cone:

\frac{36\pi}{96\pi} = \frac{36}{96} = \frac{3}{8}

Key Concept

Calculating the volume of spheres and cones, and using 2D cross-sections to find missing geometric dimensions of inscribed solids.

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