The volume of cylinder is
\(V=\pi r^2h=180.5.\)
Express h from this expresiion:
\(h=\dfrac{180.5}{\pi r^2}.\)
The full surface area of this cylinder is
\(A=2\pi r^2+2\pi rh.\)
Then in terms of r the area is
\(A(r)=2\pi r^2+2\pi r\cdot \dfrac{180.5}{\pi r^2}=2\pi r^2+\dfrac{361}{r}.\)
Find the derivative \(\dfrac{dA}{dr}:\)
\(\dfrac{dA}{dr} =4\pi r-\dfrac{361}{r^2}.\)
Equate \(\dfrac{dA}{dr}\) to zero:
\(4\pi r-\dfrac{361}{r^2}=0,\\ \\4\pi r^3=361,\\ \\ r=\sqrt[3]{\dfrac{361}{4\pi}} \approx 28.74.\)
When \(r<\sqrt[3]{\dfrac{361}{4\pi}}\), the derivative \(\dfrac{dA}{dr}\) is <0 (area is decreasing) and when \(r>\sqrt[3]{\dfrac{361}{4\pi}}\), the derivative \(\dfrac{dA}{dr}\) is >0 (area is increasing). This means that \(r=\sqrt[3]{\dfrac{361}{4\pi}}\) is point of minimum.
Thus,
\(h=\dfrac{180.5}{\pi r^2}=\dfrac{180.5}{\pi (\sqrt[3]{\dfrac{361}{4\pi}})^2}\approx 6.13.\)
Answer: the approximate values: r≈28.74 in, h≈6.13 in. and the exact values:
\(r=\sqrt[3]{\dfrac{361}{4\pi}}\) in,
\(h=\dfrac{180.5}{\pi (\sqrt[3]{\dfrac{361}{4\pi}})^2}\) in.
