If the cylinder and the cone shown have the same volume, then the radius and height of the cone could be which of these?

To determine the correct answer, we first need to calculate the volume of the cylinder.

Volume of cylinder = πr^2h
Volume of cylinder = π(5^2)(4)
Volume of cylinder = π(25)(4)
Volume of cylinder = 100π

Since the volume of the cylinder is 100π, the volume of the cone must also be 100π in order for them to have the same volume.

The formula for the volume of a cone is V = 1/3πr^2h, where r is the radius and h is the height.

100π = 1/3πr^2h
100 = 1/3 r^2h
300 = r^2h

Now, let's examine the answer choices:

1. radius = 5; height = 4
r = 5, h = 4
5^2 * 4 = 100, so this is a possible solution

2. radius = 15; height = 12
r = 15, h = 12
15^2 * 12 = 2700, this does not equal 300, so this is not a possible solution

3. radius = 15; height = 4
r = 15, h = 4
15^2 * 4 = 900, this does not equal 300, so this is not a possible solution

4. radius = 5; height = 12
r = 5, h = 12
5^2 * 12 = 300, this equals 300, so this is a possible solution

Therefore, the correct choices are:
radius of the cone = 5; height of the cone = 4
and
radius of the cone = 5; height of the cone = 12