Optimization worksheet with answers

An open box is made from a square sheet of cardboard 12 cm × 12 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.

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 330 cm³. The metal used is proportional to the total surface area. Find the radius r that uses the least material.

An open box (no lid) with a square base must hold 13500 cm³. Use the least possible material, that is the smallest area of the base plus the four sides. Find the height h.

A small firm sells its product at a fixed market price of €25 per unit. Producing q units costs C(q) = 300 + 6q + 0.1q² euros. Find the maximum profit P.

A theatre sells 340 tickets per show at €10 each. Market research shows that every €1 increase in the ticket price means 10 fewer tickets sold (and every €1 decrease, 10 more). Find the maximum revenue R per show.