Optimization worksheet with answers

An open box is made from a square sheet of cardboard 18 cm × 18 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 pen is fenced with 72 m of fencing and divided into two parts by one extra fence parallel to the sides of length x (three fences of length x and two of length y in total). Find the largest total area A that can be enclosed.

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 height h.

A rectangular plot of 1200 m² is fenced on all four sides. The side along the street, of length x, costs €20 per metre; the other three sides cost €5 per metre. Find the minimum total cost C.

A company sells q units per week. The price at which it can sell them is p = 100 − 1q euros per unit, and the cost is C(q) = 100 + 70q euros. Find the price p that maximises profit.

A theatre sells 1560 tickets per show at €25 each. Market research shows that every €1 increase in the ticket price means 40 fewer tickets sold (and every €1 decrease, 40 more). Find the maximum revenue R per show.