Optimization worksheet with answers

A farmer has 80 m of fencing to enclose a rectangular field along a straight river. No fence is needed along the river. Find the largest area A that can be enclosed.

A rectangular pen is fenced with 240 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 32000 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 750 cm³. The metal used is proportional to the total surface area. Find the height h of the can that uses the least material.

A small firm sells its product at a fixed market price of €45 per unit. Producing q units costs C(q) = 4600 + 18q + 0.02q² euros. Find the maximum profit P.

A company sells q units per week. The price at which it can sell them is p = 70 − 0.2q euros per unit, and the cost is C(q) = 2300 + 15q euros. Find the weekly output q that maximises profit.