A gardener is building a shed to protect his plants from hot weather.

Question

A gardener is building a shed to protect his plants from hot weather. Its shape is the surface of a half cylinder with radius r and length h as shown. The material needed for the two flat semi-circular ends costs 14 dollars per square foot. The material needed for the upper curved portion costs 8 dollars per square foot. A bottom will not be needed.

(a) Find the total cost, C(r)

, for building this shed as a function of the radius if the volume inside must be 400 cubic feet.

Answer 1

So the total cost will be the area x the cost per area. For this particular shed, the total area for the semi circular ends is 1/2 * pi * r² + 1/2 * pi * r² = pi * r² sq. ft. For the curved portion, the area is pi * r * h (recall that the area for the whole cylinder is 2 * pi * r * h).

The total cost of the shed is:

C = 14 * pi * r² + 8 * pi * r * h

We need it as a function of r only. Use the volume info to get h in terms of r.

The volume of a cylinder is pi * r² * h. We need half that so the second equation is:

400 = 1/2 * pi * r² * h

Solve for h: h = 800/(pi * r²)

insert this back into the first equation,

C(r) = 14 * pi * r² + 8 * pi * r * 800/(pi * r²) = 14 * pi * r² + 6400/r