Chapter 14.5, Problem 23WE

a.

To determine

To find the ratio of the surface area of the spheres.

Answer to Problem 23WE

The ratio of the surface areas is $\frac{9}{16}$ .

Explanation of Solution

Given information:

The given statement is:

“A dilation with scale factor $\frac{3}{4}$ maps a sphere with center C to a concentric sphere.”

Calculation:

The ratio of the area of the original image to its dilated image after dilation $D_{O,k}$ is $k^2$ times.

Hence, the ratio of the surface areas is $\frac{9}{16}$ .

b.

To determine

To find the ratio of the volumes of the spheres.

Answer to Problem 23WE

The ratio of the volumes is $\frac{27}{64}$ .

Explanation of Solution

Given information:

The given statement is:

“A dilation with scale factor $\frac{3}{4}$ maps a sphere with center C to a concentric sphere.”

Calculation:

The ratio of the volume of the original image to its dilated image after dilation $D_{O,k}$ is ${\left|k\right|}^3$ times.

Hence, the ratio of the volumes is $\frac{27}{64}$ .