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Chapter 10: Problem 35

Find the ratio, reduced to lowest terms, of the volume of a sphere with a radius of 3 inches to the volume of a sphere with a radius of 6 inches.

Short Answer

Expert verified
The ratio of the volume of the sphere with a radius of 3 inches to the sphere with a radius of 6 inches, reduced to lowest terms, is \( \frac{1}{8} \).

Step by step solution

01

Compute the volumes of the two spheres

Using the formula \( V = \frac{4}{3} \pi r^3 \), calculate the volume of the two spheres. For the sphere with a radius of 3 inches, the volume is \( V_1 = \frac{4}{3} \pi (3^3) \). For the sphere with radius of 6 inches, the volume is \( V_2 = \frac{4}{3} \pi (6^3) \).
02

Find the ratio between the two volumes

The ratio of volumes is obtained by dividing the volume of the smaller sphere \( V_1 \) by the larger sphere \( V_2 \). This gives \( \frac{V_1}{V_2} \)
03

Simplify the ratio

On substituting the values of \( V_1 \) and \( V_2 \) into the ratio, we get \( \frac{\frac{4}{3} \pi (3^3)}{\frac{4}{3} \pi (6^3)} \). Simplifying this, we obtain the ratio as \( \frac{1}{8} \)

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