Find the maximum volume of a rectangular box (with a lid) that can be inscribed in a cylinder of radius $30$ cm and height $60$ cm.
Consider the circle with radius $= 30$ cm. Volume of the max rectangle enclosed within it: $(\sqrt{2}r)^2 = 2r^2 = 1800$ cm$^2$. Max height $= 60$ cm (as lid on). Hence maximum volume $= (1800 \times 60)$ cm$^3$.
Find the maximum volume of a rectangular box (with a lid) that can be inscribed in a cylinder of radius $30$ cm and height $60$ cm.