11 Rakti makes open-topped cylindrical planters out of thin sheets of galvanised steel. bends a rectangle of steel to make an open cylinder and welds the joint. She She bends this cylinder to the circumference of a circular base then welds this cylinder to the circumference of a circular base.
\includegraphics[max width=\textwidth, alt={}, center]{8d9ace4b-0c15-48bd-9b0d-302f57ea9759-12_552_524_497_753} The planter must have a capacity of \(8000 \mathrm {~cm} ^ { 3 }\) Welding is time consuming, so Rakti wants the total length of weld to be a minimum.
Calculate the radius, \(r\), and height, \(h\), of a planter which requires the minimum total length of weld. Fully justify your answers, giving them to an appropriate degree of accuracy.