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 largest total area A that can be enclosed.

A rectangular plot of 300 m² is fenced on all four sides. The side along the street, of length x, costs €25 per metre; the other three sides cost €5 per metre. Find the minimum total cost C.

An open box (no lid) with a square base must hold 13500 cm³. Use the least possible material, that is the smallest area of the base plus the four sides. Find the height h.

A company sells q units per week. The price at which it can sell them is p = 70 − 0.1q euros per unit, and the cost is C(q) = 2700 + 10q euros. Find the weekly output q that maximises profit.

A theatre sells 410 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.