Optimization worksheet with answers

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

A rectangular pen is fenced with 360 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 600 m² is fenced on all four sides. The side along the street, of length x, costs €25 per metre; the other three sides cost €8 per metre. Find the length x of the street side that minimises the cost.

A rectangular plot of 300 m² is fenced on all four sides. The side along the street, of length x, costs €30 per metre; the other three sides cost €8 per metre. Find the minimum total cost C.

A small firm sells its product at a fixed market price of €75 per unit. Producing q units costs C(q) = 200 + 56q + 0.25q² euros. Find the maximum profit P.

A small firm sells its product at a fixed market price of €65 per unit. Producing q units costs C(q) = 1500 + 10q + 0.25q² euros. Find the output q that maximises profit.