A rectangular pen is fenced with 144 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 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 13500 cm³. Use the least possible material, that is the smallest area of the base plus the four sides. Find the base side x.
A rectangular plot of 600 m² is fenced on all four sides. The side along the street, of length x, costs €20 per metre; the other three sides cost €10 per metre. Find the length x of the street side that minimises the cost.
A small firm sells its product at a fixed market price of €70 per unit. Producing q units costs C(q) = 600 + 16q + 0.25q² euros. Find the maximum profit P.
A theatre sells 1650 tickets per show at €15 each. Market research shows that every €1 increase in the ticket price means 50 fewer tickets sold (and every €1 decrease, 50 more). Find the maximum revenue R per show.
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