Optimization worksheet with answers

An open box is made from a rectangular sheet of cardboard 30 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 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 largest possible volume V.

A rectangular plot of 200 m² is fenced on all four sides. The side along the street, of length x, costs €25 per metre; the other three sides cost €10 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 height h.

A small firm sells its product at a fixed market price of €20 per unit. Producing q units costs C(q) = 100 + 10q + 0.1q² euros. Find the output q that maximises profit.

A company sells q units per week. The price at which it can sell them is p = 70 − 0.25q euros per unit, and the cost is C(q) = 1500 + 15q euros. Find the price p that maximises profit.