Optimization worksheet with answers

An open box is made from a square sheet of cardboard 48 cm × 48 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 rectangular sheet of cardboard 9 cm × 24 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 rectangular plot of 300 m² is fenced on all four sides. The side along the street, of length x, costs €40 per metre; the other three sides cost €12 per metre. Find the minimum total cost C.

An open box (no lid) with a square base must hold 62500 cm³. Use the least possible material, that is the smallest area of the base plus the four sides. Find the base side x.

A theatre sells 290 tickets per show at €15 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.

A small firm sells its product at a fixed market price of €40 per unit. Producing q units costs C(q) = 200 + 22q + 0.2q² euros. Find the maximum profit P.