Express the height of a cylinder in terms of the radius if the cylinder and a sphere have the same volume and radius.

To find the height of the cylinder in terms of the radius when both the cylinder and the sphere have the same volume and radius, we start with the formulas for the volumes of both shapes.

The volume ( V ) of a sphere is given by the formula:

[

V_{\text{sphere}} = \frac{4}{3} \pi r^3

]

The volume ( V ) of a cylinder is given by the formula:

[

V_{\text{cylinder}} = \pi r^2 h

]

Since we know both volumes are equal, we can set them equal to each other:

[

\frac{4}{3} \pi r^3 = \pi r^2 h

]

To simplify, we can divide both sides by ( \pi r^2 ) (assuming ( r \neq 0 )):

[

\frac{4}{3} r = h

]

This equation shows that the height ( h ) of the cylinder is (\frac{4}{3}) times the radius ( r ). Therefore, expressing this relationship, we have:

[

h = \frac{4}{3} r

]

This