O-Level Differentiation and integration: worked solution

Question

For that open tank (base 4l by l, height h, outer surface area 5 m², volume V = (2/5)(5l − 4l³)), find the base area that lets it hold the most water when filled to the brim.

Given V = (2/5)(5l - 4l³) = 2l - (8/5)l³. dV/dl = 2 - (24/5)l². Set dV/dl = 0: (24/5)l² = 2 => l² = 10/24 = 5/12. d^2V/dl² = -(48/5)l < 0, so this gives a maximum volume. Base area = 4l x l = 4l² = 4(5/12) = 5/3 m² ≈ 1.67 m².