Find the ratio of the radius r to the height h of a cylinder which maximizes its total surface area subject to the constraint that its volume is constant.

Question

Accepted Answer

The volume V = πr²h and area A = 2πrh + 2πr², so we consider the function F given by

F = A + λV, (C.92)

and solve

[latex]\frac{∂F }{∂h}[/latex] = 2πr + λπr² = 0 , (C.93)

[latex]\frac{∂F }{∂r}[/latex] = 2πh + 4πr + 2λπrh = 0 , (C.94)

which yields λ = −2/r and hence h = 2 r.

Question C.2: Find the ratio of the radius r to the height h of a cylinder......