A cylindrical biscuit tin has a close-fitting lid which overlaps the tin by 1 cm, as shown. The radii of the tin and the lid are both x cm. The tin and the lid are made from a thin sheet of metal of area 80π cm² and there is no wastage. The volume of the tin is V cm³.

a. Show that V = π(40x − x² − x³). Given that x can vary:

b. use differentiation to find the positive value of x for which V is stationary.

c. Prove that this value of x gives a maximum value of V.

d. Find this maximum value of V.

e. Determine the percentage of the sheet metal used in the lid when V is a maximum.

Lid Tin x cm x cm 1 cm