Question

Find the minimum amount of tin sheet in square inches that can be made into a closed cylinder having a volume of 108 cubic inches

Answer 1

$V=\pi R^2L=108$

$L=108/\pi R^2$

$S=2\pi RL+2\pi R^2=216/R+2\pi R^2$

$\frac{dS}{dR}=-216/R^2+4\pi R=0$

$4\pi R^3-216=0$

$R^3=216/(4\pi)$

$R_{min}=2.58\ in$

$S_{min}=216/2.58+2\pi\cdot2.58^2=125.53\ in^2$