You have been asked to design a one litre can shaped like a right circular cylinder what dimensions

Question

You have been asked to design a one litre can shaped like a right circular cylinder what dimensions will use the least material. What is the height in cm rounded off to the nearest integer?

Professor PC Thomas Classes · Answer

Surface area $=2 \pi r^2+2 \pi r h$ For a first approximation, we can ignore the thickness and waste of material. $$ \begin{aligned} & 	herefore 	ext { Volume }=\pi r^2 h=100 m l \Rightarrow h=\frac{1000}{\pi r^2} \ & A=2 \pi r^2+2 \pi r h \end{aligned} $$ $\begin{aligned} & =2 \pi r^2+2 \pi r\left(\frac{1000}{\pi r^2}ight)=2 \pi r^2+\frac{200}{r} \ & A^{\prime}=4 \pi r-\frac{2000}{r^2}=0 \ & 2000=4 \pi r^3 \ & r=\left(\frac{500}{\pi}ight)^{1 / 3} \frac{d^2 A}{d r^2}>0 \ & \Rightarrow h=\frac{1000}{\pi r^2}=2 \sqrt[3]{\frac{500}{\pi}}=2 r \ & 	herefore r \cong 5.42 \mathrm{~cm}, h \cong 10.84 \mathrm{~cm}\end{aligned}$