Optimization worksheet with answers

A farmer has 40 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 120 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.

A rectangular plot of 200 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 rectangular plot of 200 m² is fenced on all four sides. The side along the street, of length x, costs €30 per metre; the other three sides cost €12 per metre. Find the minimum total cost C.

A theatre sells 260 tickets per show at €12 each. Market research shows that every €1 increase in the ticket price means 10 fewer tickets sold (and every €1 decrease, 10 more). Find the ticket price p that maximises revenue.

A small firm sells its product at a fixed market price of €20 per unit. Producing q units costs C(q) = 3200 + 4q + 0.01q² euros. Find the maximum profit P.