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 length x of each side perpendicular to the river for the largest area.

An open box is made from a rectangular sheet of cardboard 18 cm × 48 cm by cutting equal squares of side x from the four corners and folding up the sides. Find the largest possible volume V.

A rectangular plot of 600 m² is fenced on all four sides. The side along the street, of length x, costs €15 per metre; the other three sides cost €10 per metre. Find the minimum total cost C.

A rectangular plot of 800 m² is fenced on all four sides. The side along the street, of length x, costs €20 per metre; the other three sides cost €12 per metre. Find the minimum total cost C.

A company sells q units per week. The price at which it can sell them is p = 100 − 1q euros per unit, and the cost is C(q) = 200 + 55q euros. Find the maximum weekly profit P.

A theatre sells 310 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.