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 length y of the side parallel to the river for the largest area.

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.

A rectangular plot of 600 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 1200 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 750 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 company sells q units per week. The price at which it can sell them is p = 190 − 0.1q euros per unit, and the cost is C(q) = 19500 + 65q euros. Find the price p that maximises profit.