A farmer's silo is the shape of a cylinder with a hemisphere as the roof.

Question

A farmer's silo is the shape of a cylinder with a hemisphere as the roof. If the radius of the hemisphere is 10 feet and the height of the silo is h feet, express the volume of the silo as a function of h.

A. V(h) = 100 πh +  πh²

B. V(h) = 100 π(h² - 10) +  π

C. V(h) = 4100 π(h - 10) +  π

D. V(h) = 100 π(h - 10) +  π

Answer 1

The answer is D: V(h) = 100 π(h - 10) +  π

First, Split the silo up into the cylinder and the hemisphere

The volume of a cylinder is π*height*radius²

The cylinder of the silo has a height of h-10, because the hemisphere has a "height" of 10 as that is the radius of the cylinder

The volume of the silo's cylinder:

V(SiloCylinder) = π*height*radius² = π*(h-10)*10² = 100π(h-10)

The volume of a hemisphere is half of the volume of a sphere = 0.5 * 4/3 * π * r³

Volume of the silo's hemisphere = 0.5 * 4/3 * π * 10³ = 2000/3*π

Now we add the two to get the volume of the entire silo

V(Silo) = 100π(h-10) + 2000/3*π