Optimization worksheet with answers

An open box is made from a square sheet of cardboard 12 cm × 12 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 pen is fenced with 360 m of fencing and divided into two parts by one extra fence parallel to the sides of length x (three fences of length x and two of length y in total). Find the length x that gives the largest total area.

An open box (no lid) with a square base must hold 32000 cm³. Use the least possible material, that is the smallest area of the base plus the four sides. Find the height h.

An open box (no lid) with a square base must hold 4000 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 company sells q units per week. The price at which it can sell them is p = 150 − 0.2q euros per unit, and the cost is C(q) = 5900 + 25q euros. Find the weekly output q that maximises profit.

A small firm sells its product at a fixed market price of €80 per unit. Producing q units costs C(q) = 2700 + 20q + 0.2q² euros. Find the output q that maximises profit.