Answer: \(S=\frac53\pi x^2+\frac{1000}{x}\), minimum when \(x=\sqrt[3]{\frac{300}{\pi}}\approx4.57\).

(a) The volume of the object is the volume of a hemisphere plus the volume of a cylinder:

\(\frac23\pi x^3+\pi x^2y=500.\)

Rearrange to find \(y\):

\(\pi x^2y=500-\frac23\pi x^3.\)

So

\(y=\frac{500}{\pi x^2}-\frac23x.\)

The surface area is the curved area of the hemisphere, plus the curved area of the cylinder, plus the circular base:

\(S=2\pi x^2+2\pi xy+\pi x^2.\)

Substitute the expression for \(y\):

Therefore

\(S=3\pi x^2+\frac{1000}{x}-\frac43\pi x^2.\)

Hence

\(S=\frac53\pi x^2+\frac{1000}{x}.\)

(b) Differentiate:

At a minimum,

\(\frac{10}{3}\pi x-\frac{1000}{x^2}=0.\)

So

\(\frac{10}{3}\pi x^3=1000.\)

Thus

\(x^3=\frac{300}{\pi}.\)

Therefore

\(x=\sqrt[3]{\frac{300}{\pi}}\approx4.57.\)