Optimization worksheet with answers

A farmer has 320 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 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 y of the side parallel to the river for the largest 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 height h.

A rectangular plot of 300 m² is fenced on all four sides. The side along the street, of length x, costs €40 per metre; the other three sides cost €8 per metre. Find the minimum total cost C.

A theatre sells 1100 tickets per show at €15 each. Market research shows that every €1 increase in the ticket price means 50 fewer tickets sold (and every €1 decrease, 50 more). Find the ticket price p that maximises revenue.

A small firm sells its product at a fixed market price of €50 per unit. Producing q units costs C(q) = 3200 + 32q + 0.01q² euros. Find the output q that maximises profit.