Optimization worksheet with answers

An open box is made from a square sheet of cardboard 30 cm × 30 cm by cutting equal squares of side x from the four corners and folding up the sides. Find the largest possible volume V.

An open box is made from a square sheet of cardboard 42 cm × 42 cm by cutting equal squares of side x from the four corners and folding up the sides. Find the cut x that gives the largest volume.

A closed cylindrical can (with a top and a bottom) must hold 250 cm³. The metal used is proportional to the total surface area. Find the smallest possible surface area A.

An open box (no lid) with a square base must hold 32000 cm³. Use the least possible material, that is the smallest area of the base plus the four sides. Find the minimum surface area S.

A small firm sells its product at a fixed market price of €70 per unit. Producing q units costs C(q) = 100 + 52q + 0.5q² euros. Find the maximum profit P.

A theatre sells 875 tickets per show at €15 each. Market research shows that every €1 increase in the ticket price means 25 fewer tickets sold (and every €1 decrease, 25 more). Find the ticket price p that maximises revenue.