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.
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 y of the side parallel 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 €40 per metre; the other three sides cost €8 per metre. Find the length x of the street side that minimises the cost.
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 base side x.
A small firm sells its product at a fixed market price of €40 per unit. Producing q units costs C(q) = 100 + 26q + 0.1q² euros. Find the maximum profit P.
A company sells q units per week. The price at which it can sell them is p = 190 − 0.5q euros per unit, and the cost is C(q) = 200 + 150q euros. Find the maximum weekly profit P.
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