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