Optimization worksheet with answers

A farmer has 240 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 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 plot of 1200 m² is fenced on all four sides. The side along the street, of length x, costs €30 per metre; the other three sides cost €8 per metre. Find the length x of the street side that minimises the cost.

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 theatre sells 1560 tickets per show at €15 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.

A small firm sells its product at a fixed market price of €40 per unit. Producing q units costs C(q) = 900 + 6q + 0.2q² euros. Find the output q that maximises profit.