Optimization worksheet with answers

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 farmer has 40 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.

An open box (no lid) with a square base must hold 62500 cm³. Use the least possible material, that is the smallest area of the base plus the four sides. Find the minimum surface area S.

A closed cylindrical can (with a top and a bottom) must hold 1000 cm³. The metal used is proportional to the total surface area. Find the radius r that uses the least material.

A company sells q units per week. The price at which it can sell them is p = 200 − 1q euros per unit, and the cost is C(q) = 1600 + 55q euros. Find the price p that maximises profit.

A small firm sells its product at a fixed market price of €75 per unit. Producing q units costs C(q) = 5200 + 24q + 0.05q² euros. Find the output q that maximises profit.