Question

The volume of the cone is 25π.

Match the Question with the correct answer.

(5 points)
How many times larger is the volume of the sphere than the volume of the cone IF the cone and the sphere had the same dimensions?
How many times larger is the volume of the cylinder than the volume of the cone IF the cone and the cylinder had the same dimensions?
The volume of the sphere if the radius is the same value as the cone.
The volume of the cylinder
In order for the volume of the cone + the volume of the sphere = the volume of the cylinder, the height would have to be _______, not 3r for the cone and cylinder.






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Answer 1

Assuming the cone and cylinder were originally taken with height h = 3r (so (1/3)πr^2(3r) = πr^3 = 25π ⇒ r^3 = 25):

1) How many times larger is the volume of the sphere than the volume of the cone IF the cone and the sphere had the same dimensions?
- 4/3 times larger. (V_sphere / V_cone = (4/3)πr^3 / πr^3 = 4/3.)

2) How many times larger is the volume of the cylinder than the volume of the cone IF the cone and the cylinder had the same dimensions?
- 3 times larger. (V_cyl / V_cone = πr^2h / (1/3)πr^2h = 3.)

3) The volume of the sphere if the radius is the same value as the cone.
- V_sphere = (4/3)πr^3 = (4/3)π·25 = (100/3)π.

4) The volume of the cylinder.
- V_cylinder = 3·V_cone = 3·25π = 75π (with h = 3r).

5) In order for the volume of the cone + the volume of the sphere = the volume of the cylinder, the height would have to be _______, not 3r for the cone and cylinder.
- h = 2r. (Solve (1/3)πr^2h + (4/3)πr^3 = πr^2h ⇒ h = 2r.)