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 farmer has 400 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 closed cylindrical can (with a top and a bottom) must hold 500 cm³. The metal used is proportional to the total surface area. Find the smallest possible surface area A.

A rectangular plot of 200 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 small firm sells its product at a fixed market price of €55 per unit. Producing q units costs C(q) = 1100 + 8q + 0.25q² euros. Find the maximum profit P.

A company sells q units per week. The price at which it can sell them is p = 120 − 0.2q euros per unit, and the cost is C(q) = 3000 + 10q euros. Find the weekly output q that maximises profit.