Optimization worksheet with answers

A farmer has 320 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 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 rectangular plot of 400 m² is fenced on all four sides. The side along the street, of length x, costs €20 per metre; the other three sides cost €8 per metre. Find the minimum total cost C.

A closed cylindrical can (with a top and a bottom) must hold 400 cm³. The metal used is proportional to the total surface area. Find the height h of the can that uses the least material.

A company sells q units per week. The price at which it can sell them is p = 60 − 0.2q euros per unit, and the cost is C(q) = 1900 + 10q euros. Find the weekly output q that maximises profit.

A company sells q units per week. The price at which it can sell them is p = 170 − 0.25q euros per unit, and the cost is C(q) = 3400 + 40q euros. Find the price p that maximises profit.