Max volume of a cylinder

Question

Find maximum volume of a cylinder of which the sum of height and the circumference of the base does not exceed 108 cm. How to solve this? Precisely what is the expression that should be minimized? How to minimize it properly?

Answer 1

The volume of a cylinder is $\pi r^{2} h$. In your question, you have the constraint $2\pi r+h \leq 108$. It is obvious that the cylinder you seek will satisfy $2\pi r+h=108$. Hence you want to maximize $\pi r^{2} (108-2\pi r)$ for $r \geq 0$.