Find the ratio, reduced to lowest terms, of the volume of a sphere with a radius of 2 inches to the volume of a sphere with a radius of 6 inches. The ratio is (Type an integer or a simplified fraction.)
Solution
Step 1 ：Calculate the volume of the sphere with a radius of 2 inches using the formula \(V = \frac{4}{3}\pi r^3\). Substitute \(r = 2\) into the formula to get \(V1 = \frac{4}{3}\pi(2)^3 = \frac{4}{3}\pi(8) = \frac{32}{3}\pi\) cubic inches.
Step 2 ：Calculate the volume of the sphere with a radius of 6 inches using the formula \(V = \frac{4}{3}\pi r^3\). Substitute \(r = 6\) into the formula to get \(V2 = \frac{4}{3}\pi(6)^3 = \frac{4}{3}\pi(216) = 288\pi\) cubic inches.
Step 3 ：Calculate the ratio of the volume of the first sphere to the volume of the second sphere using the formula \(\frac{V1}{V2}\). Substitute the values of \(V1\) and \(V2\) into the formula to get \(\frac{V1}{V2} = \frac{\frac{32}{3}\pi}{288\pi} = \frac{32}{3} / 288 = \frac{32}{864} = \frac{1}{27}\).
Step 4 ：\(\boxed{\frac{1}{27}}\) is the ratio of the volume of a sphere with a radius of 2 inches to the volume of a sphere with a radius of 6 inches.
