Question

A farmer wants to build a new grain silo. the shape of the silo is to be a cylinder with a hemisphere on top, where the radius of the hemisphere is to be the same length as the radius of the base of the cylinder. the farmer would like the height of the silo's cylinder portion to be 3 times the diameter of the base of the cylinder. what should the radius of the silo be if the silo is to hold 22500pi cubic feet of grain?

Can someone explain how to do this but like dumb it down some?

Answer 1

15 ft

Explanation:

The shape of the silo is a cylinder with a hemisphere (half a sphere) on top. The radius of the hemisphere is the same as the radius of the cylinder. The height of the cylinder is 3 times its diameter. Remember that diameter is twice the radius.

Volume of a hemisphere is:

V = ⅔ π r³

Volume of a cylinder is:

V = π r² h

The total volume is therefore:

V = ⅔ π r³ + π r² h

We know that V = 22500π ft³, and h = 3d = 6r. Plugging this into the equation:

22500π ft³ = ⅔ π r³ + π r² (6r)

22500π ft³ = ⅔ π r³ + 6π r³

Divide both sides by π:

22500 ft³ = ⅔ r³ + 6 r³

Multiply both sides by 3:

67500 ft³ = 2 r³ + 18 r³

Combine like terms:

67500 ft³ = 20 r³

Divide both sides by 20:

3375 ft³ = r³

Take the cube root:

r = 15 ft

The radius of the silo must be 15 feet.