Question

a sphere and a cylinder have the same radius and height. the volume of the cylinder is 48 cm³. what is the volume of the sphere? cm³

Explanation:

Step1: Write the volume formula for cylinder

The volume formula for a cylinder is \(V_{cylinder}=\pi r^{2}h\). Given \(V_{cylinder} = 48\space cm^{3}\), so \(\pi r^{2}h=48\).

Step2: Relate the height of the sphere and cylinder

Since the sphere and cylinder have the same radius and height, for a sphere \(h = 2r\) (diameter).

Step3: Write the volume formula for sphere

The volume formula for a sphere is \(V_{sphere}=\frac{4}{3}\pi r^{3}\). From \(h = 2r\) and \(\pi r^{2}h = 48\), substitute \(h = 2r\) into \(\pi r^{2}h\): \(\pi r^{2}(2r)=48\), which simplifies to \(2\pi r^{3}=48\), then \(\pi r^{3}=24\).

Step4: Calculate the volume of the sphere

Substitute \(\pi r^{3}=24\) into \(V_{sphere}=\frac{4}{3}\pi r^{3}\). So \(V_{sphere}=\frac{4}{3}\times24\).

Answer:

\(32\)