Optimization worksheet with answers

A farmer has 400 m of fencing to enclose a rectangular field along a straight river. No fence is needed along the river. Find the length x of each side perpendicular to the river for the largest area.

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.

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.

An open box (no lid) with a square base must hold 32000 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 small firm sells its product at a fixed market price of €55 per unit. Producing q units costs C(q) = 100 + 38q + 0.25q² euros. Find the maximum profit P.

A small firm sells its product at a fixed market price of €20 per unit. Producing q units costs C(q) = 100 + 6q + 0.5q² euros. Find the output q that maximises profit.