Optimization worksheet with answers

A farmer has 200 m of fencing to enclose a rectangular field along a straight river. No fence is needed along the river. Find the largest area A that can be enclosed.

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.

A rectangular plot of 450 m² is fenced on all four sides. The side along the street, of length x, costs €40 per metre; the other three sides cost €5 per metre. Find the minimum total cost C.

An open box (no lid) with a square base must hold 13500 cm³. Use the least possible material, that is the smallest area of the base plus the four sides. Find the base side x.

A theatre sells 650 tickets per show at €12 each. Market research shows that every €1 increase in the ticket price means 25 fewer tickets sold (and every €1 decrease, 25 more). Find the maximum revenue R per show.

A small firm sells its product at a fixed market price of €45 per unit. Producing q units costs C(q) = 100 + 36q + 0.2q² euros. Find the output q that maximises profit.