Optimization worksheet with answers

An open box is made from a square sheet of cardboard 24 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.

An open box is made from a rectangular sheet of cardboard 14 cm × 30 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 450 m² is fenced on all four sides. The side along the street, of length x, costs €40 per metre; the other three sides cost €8 per metre. Find the length x of the street side that minimises the cost.

A rectangular plot of 400 m² is fenced on all four sides. The side along the street, of length x, costs €25 per metre; the other three sides cost €8 per metre. Find the minimum total cost C.

A theatre sells 750 tickets per show at €10 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 maximum revenue R per show.

A company sells q units per week. The price at which it can sell them is p = 150 − 0.25q euros per unit, and the cost is C(q) = 1500 + 80q euros. Find the price p that maximises profit.