Optimization worksheet with answers

An open box is made from a rectangular sheet of cardboard 21 cm × 45 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.

An open box is made from a square sheet of cardboard 42 cm × 42 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 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 €12 per metre. Find the minimum total cost C.

A closed cylindrical can (with a top and a bottom) must hold 750 cm³. The metal used is proportional to the total surface area. Find the radius r that uses the least material.

A small firm sells its product at a fixed market price of €50 per unit. Producing q units costs C(q) = 200 + 32q + 0.2q² 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) = 16200 + 18q + 0.01q² euros. Find the maximum profit P.