A sphere and a cone have the same volume. Each figure has a radius of 3 inches. What is the height of the cone?

If r = 3

Volume of sphere = (4/3)π r^3 = 36π
volume of a cone = (1/3)π r^2 h =3π h

3π h = 36π
3h = 36
h = 12

To find the height of the cone, we need to first equate the volumes of the sphere and the cone. The volume of a sphere is given by the formula V_sphere = (4/3) * π * r^3, where r is the radius. The volume of a cone is given by the formula V_cone = (1/3) * π * r^2 * h, where h is the height.

Given that the radius of both the sphere and the cone is 3 inches, we can write:

V_sphere = (4/3) * π * 3^3
V_cone = (1/3) * π * 3^2 * h

Since we know that the volumes of the sphere and the cone are equal, we can set up the equation:

(4/3) * π * 3^3 = (1/3) * π * 3^2 * h

Now, we can simplify the equation by canceling out common factors:

4 * 3 * 3 = 1 * 3^2 * h

36 = 9 * h

To isolate h, we divide both sides of the equation by 9:

h = 36 / 9

Simplifying further:

h = 4

Therefore, the height of the cone is 4 inches.