A rectangular pen is fenced with 120 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.
A farmer has 320 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 plot of 1200 m² is fenced on all four sides. The side along the street, of length x, costs €30 per metre; the other three sides cost €5 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 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 €20 per unit. Producing q units costs C(q) = 100 + 10q + 0.2q² euros. Find the maximum profit P.
A company sells q units per week. The price at which it can sell them is p = 150 − 0.1q euros per unit, and the cost is C(q) = 9100 + 15q euros. Find the price p that maximises profit.
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