Question

If a cone has the same radius and height as a cylinder, the volume of the cone is the volume of the cylinder. If a cylinder and a sphere have the same radius and the cylinder’s height is twice its radius, then the volume of the sphere is the volume of the cylinder

Answer 1

We are given a cone that has the same radius and height as a cylinder. The volume of a cone is given by:

$V_c=\frac{1}{3}\pi r^2h$

The volume of a cylinder is:

$V_c=\pi r^2h$

Therefore, the volume of the cone is one-third the volume of the cylinder.

We are also told that a cylinder and a sphere have the same radius and the cylinder’s height is twice its radius.

The volume of a sphere is:

$V_{sp}=\frac{4}{3}\pi r^3$

The volume of a cylinder is:

$V_c=\frac{1}{3}\pi r^2h$

We have that:

$h=2r$

Replacing in the formula for the volume of the cylinder:

$V_c=\frac{1}{3}\pi r^2(2r)$

Simplifying:

$V_c=\frac{2}{3}\pi r^3$

Multiplying by 2:

$2V_c=\frac{4}{3}\pi r^3$

Therefore:

$2V_c=V_{sp}$

Therefore, the volume of the sphere is twice the volume of the cylinder.