The volume of a cone varies jointly as the square of its radius and its height. If the volume of a cone is 25π cubic inches when the radius is 5 inches and the height is 3 inches, find the volume of a cone when the radius is 3 inches and the height is 6 inches

v = k r^2 h

That is, v/(r^2 h) is constant
So, you want v such that
v/(3^2 * 6) = 25π/(5^2 * 3)

or, knowing as you do that
v = π/3 r^2 h
just plug in your numbers directly

To find the volume of the cone when the radius is 3 inches and the height is 6 inches, we can use the concept of joint variation and the given information.

First, let's identify the variables involved in the problem:
- Let V represent the volume of the cone.
- Let r represent the radius of the cone.
- Let h represent the height of the cone.

According to the problem statement, the volume V varies jointly as the square of the radius (r^2) and the height (h). This can be written as:
V = k * (r^2) * h

Given that the volume of the cone is 25π cubic inches when the radius is 5 inches and the height is 3 inches, we have:
25π = k * (5^2) * 3
25π = 75k

To find the value of k, we can divide both sides of the equation by 75:
k = (25π) / 75
k = π / 3

Now that we have the value of k, we can substitute it back into the equation and solve for the volume of the cone when the radius is 3 inches and the height is 6 inches:
V = (π / 3) * (3^2) * 6
V = (π / 3) * 9 * 6
V = (π / 3) * 54
V = 18π

Therefore, the volume of the cone when the radius is 3 inches and the height is 6 inches is 18π cubic inches.