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 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 (no lid) with a square base must hold 108000 cm³. Use the least possible material, that is the smallest area of the base plus the four sides. Find the minimum surface area S.

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 minimum surface area S.

A company sells q units per week. The price at which it can sell them is p = 110 − 0.25q euros per unit, and the cost is C(q) = 2200 + 25q euros. Find the price p that maximises profit.

A small firm sells its product at a fixed market price of €20 per unit. Producing q units costs C(q) = 1800 + 8q + 0.01q² euros. Find the maximum profit P.