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.

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 largest total area A that can be enclosed.

A closed cylindrical can (with a top and a bottom) must hold 330 cm³. The metal used is proportional to the total surface area. Find the height h of the can that uses the least material.

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 €8 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 = 190 − 0.2q euros per unit, and the cost is C(q) = 11500 + 15q euros. Find the price p that maximises profit.

A theatre sells 290 tickets per show at €15 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.