Optimization worksheet with answers

An open box is made from a square sheet of cardboard 30 cm × 30 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 largest total area A that can be enclosed.

An open box (no lid) with a square base must hold 108000 cm³. Use the least possible material, that is the smallest area of the base plus the four sides. Find the minimum surface area S.

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 base side x.

A small firm sells its product at a fixed market price of €80 per unit. Producing q units costs C(q) = 5800 + 26q + 0.05q² euros. Find the maximum profit P.

A small firm sells its product at a fixed market price of €45 per unit. Producing q units costs C(q) = 2000 + 22q + 0.02q² euros. Find the maximum profit P.