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.
A farmer has 160 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.
A rectangular plot of 400 m² is fenced on all four sides. The side along the street, of length x, costs €20 per metre; the other three sides cost €8 per metre. Find the length x of the street side that minimises the cost.
A closed cylindrical can (with a top and a bottom) must hold 400 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 − 2q euros per unit, and the cost is C(q) = 600 + 90q euros. Find the weekly output q that maximises profit.
A small firm sells its product at a fixed market price of €55 per unit. Producing q units costs C(q) = 100 + 38q + 0.2q² euros. Find the output q that maximises profit.
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