Optimization worksheet with answers

A farmer has 160 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.

An open box is made from a rectangular sheet of cardboard 9 cm × 24 cm by cutting equal squares of side x from the four corners and folding up the sides. Find the cut x that gives the largest volume.

A rectangular plot of 200 m² is fenced on all four sides. The side along the street, of length x, costs €40 per metre; the other three sides cost €10 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 €20 per metre; the other three sides cost €5 per metre. Find the length x of the street side that minimises the cost.

A small firm sells its product at a fixed market price of €40 per unit. Producing q units costs C(q) = 400 + 8q + 0.25q² euros. Find the output q that maximises profit.

A small firm sells its product at a fixed market price of €75 per unit. Producing q units costs C(q) = 200 + 56q + 0.2q² euros. Find the output q that maximises profit.