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Chapter 10: Problem 36

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 9 inches.

Short Answer

Expert verified
The ratio of the volume of a sphere with a radius of 3 inches to the volume of a sphere with a radius of 9 inches, reduced to its simplest form, is \[ \frac{1}{27} \]

Step by step solution

01

Calculate the volume of the first sphere

We use the formula \[ V = \frac{4}{3}\pi r^3 \] to calculate the volume of the first sphere. Here, the radius r is 3 inches. So, \[ V1 = \frac{4}{3}\pi (3)^3 \] which simplifies to \[ V1 = 36\pi \] cubic inches.
02

Calculate the volume of the second sphere

Next, use the volume formula again to calculate the volume of the second sphere, where the radius r is 9 inches. So, we have \[ V2 = \frac{4}{3}\pi (9)^3 \] which simplifies to \[ V2 = 972\pi \] cubic inches.
03

Calculate the ratio of the volumes

To find the ratio of the volumes of the two spheres we divide the volume of the first sphere by the volume of the second sphere, as follows: \[ ratio = \frac{V1}{V2} = \frac{36\pi}{972\pi} \]. The \(\pi\) cancels out, and simplifying the resulting fraction of \[ \frac{36}{972} \] gives us \[ \frac{1}{27} \].

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