Optimization worksheet with answers

A farmer has 240 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 48 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.

An open box (no lid) with a square base must hold 108000 cm³. Use the least possible material, that is the smallest area of the base plus the four sides. Find the minimum surface area S.

A rectangular plot of 400 m² is fenced on all four sides. The side along the street, of length x, costs €25 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 = 150 − 2q euros per unit, and the cost is C(q) = 200 + 100q euros. Find the price p that maximises profit.

A small firm sells its product at a fixed market price of €55 per unit. Producing q units costs C(q) = 7200 + 24q + 0.01q² euros. Find the output q that maximises profit.