Question

a sphere and a cylinder have the same radius and height. the volume of the cylinder is 50 ft³. what is the volume of the sphere? 100 ft³, 50/3 ft³, 75 ft³, 100/3 ft³

Explanation:

Step1: Recall the volume formulas

The volume formula of a cylinder is \(V_{cylinder}=\pi r^{2}h\), and the volume formula of a sphere is \(V_{sphere}=\frac{4}{3}\pi r^{3}\). Given that \(h = 2r\) (for a sphere and a cylinder with the same radius and height, the height of the cylinder is equal to the diameter of the sphere), and \(V_{cylinder}=\pi r^{2}h = 50\). Substitute \(h = 2r\) into the cylinder - volume formula: \(V_{cylinder}=\pi r^{2}(2r)=2\pi r^{3}=50\).

Step2: Solve for \(\pi r^{3}\)

From \(2\pi r^{3}=50\), we can get \(\pi r^{3}=25\).

Step3: Calculate the volume of the sphere

Substitute \(\pi r^{3}=25\) into the sphere - volume formula \(V_{sphere}=\frac{4}{3}\pi r^{3}\). Then \(V_{sphere}=\frac{4}{3}\times25=\frac{100}{3}\).

Answer:

\(\frac{100}{3}\text{ ft}^3\)