Question

a sphere and a cylinder have the same radius and height. the volume of the cylinder is 21 m³. what is the volume of the sphere? 6 m³ 14 m³ 7 m³ 28 m³

Explanation:

Step1: Recall the volume formula for a cylinder

The volume formula for a cylinder is \( V_{cylinder}=\pi r^{2}h\). Given \( V_{cylinder} = 21m^{3}\).

Step2: Recall the relationship between the sphere and the cylinder

For a sphere and a cylinder with the same radius \( r\), and the height \( h\) of the cylinder equal to the diameter \( d = 2r\) of the sphere.
Substitute \( h = 2r\) into the cylinder - volume formula: \( V_{cylinder}=\pi r^{2}(2r)=2\pi r^{3}\). So, \( 2\pi r^{3}=21\).

Step3: Recall the volume formula for a sphere

The volume formula for a sphere is \( V_{sphere}=\frac{4}{3}\pi r^{3}\).
From \( 2\pi r^{3}=21\), we can find \(\pi r^{3}=\frac{21}{2}\).
Then substitute \(\pi r^{3}\) into the sphere - volume formula: \( V_{sphere}=\frac{4}{3}\times\frac{21}{2}\).
Calculate \(\frac{4}{3}\times\frac{21}{2}=\frac{4\times21}{3\times2}=\frac{84}{6} = 14\).

Answer:

\(14m^{3}\)