Question

A metal storage tank with volume 5m^3 is to be constructed in the shape of a right circular cylinder surmounted by a hemisphere. What dimensions will require the least amount of metal? (Hint: for atleast amount of metal, the surface area should be a min)

Answer 1

\[V = \pi h r^{2}+\frac{2}{3}\pi r^{3} = 5\]
\[SA = \pi r^{2}+2\pi h r+2\pi r^{2}\]
From volume equation solve for h
\[h = \frac{15-2\pi r^{3}}{3\pi r^{2}}\]
substitute into surface area equation
\[SA = 3\pi r^{2}+\frac{2\pi(15-2\pi r^{3})}{3\pi r}\]
differentiate and set equal to 0
\[\frac{dA}{dr} = 6\pi r + \frac{-36(\pi r)^{3}-6\pi^{2}(15-2\pi r^{3})}{(3\pi r)^{2}}=0\]
\[\rightarrow 54 \pi^{3} r^{3} = 24\pi^{3} r^{3}+90\pi^{2}\]
\[\rightarrow r^{3} = \frac{3}{\pi}\]
\[r \approx 0.985\]