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 largest area A that can be enclosed.

A rectangular pen is fenced with 96 m of fencing and divided into two parts by one extra fence parallel to the sides of length x (three fences of length x and two of length y in total). Find the length x that gives the largest total area.

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.

A rectangular plot of 800 m² is fenced on all four sides. The side along the street, of length x, costs €25 per metre; the other three sides cost €5 per metre. Find the minimum total cost C.

A small firm sells its product at a fixed market price of €25 per unit. Producing q units costs C(q) = 100 + 12q + 0.5q² euros. Find the maximum profit P.

A theatre sells 1075 tickets per show at €25 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 ticket price p that maximises revenue.