Optimization worksheet with answers

An open box is made from a square sheet of cardboard 24 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 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 closed cylindrical can (with a top and a bottom) must hold 250 cm³. The metal used is proportional to the total surface area. Find the height h of the can that uses the least material.

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 €12 per metre. Find the minimum total cost C.

A small firm sells its product at a fixed market price of €50 per unit. Producing q units costs C(q) = 1300 + 8q + 0.2q² euros. Find the maximum profit P.

A small firm sells its product at a fixed market price of €40 per unit. Producing q units costs C(q) = 100 + 28q + 0.25q² euros. Find the maximum profit P.