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 length x that gives the largest total area.
A farmer has 120 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 closed cylindrical can (with a top and a bottom) must hold 1000 cm³. The metal used is proportional to the total surface area. Find the radius r that uses the least material.
A rectangular plot of 200 m² is fenced on all four sides. The side along the street, of length x, costs €25 per metre; the other three sides cost €8 per metre. Find the length x of the street side that minimises the cost.
A theatre sells 190 tickets per show at €10 each. Market research shows that every €1 increase in the ticket price means 10 fewer tickets sold (and every €1 decrease, 10 more). Find the maximum revenue R per show.
A small firm sells its product at a fixed market price of €70 per unit. Producing q units costs C(q) = 4800 + 26q + 0.05q² euros. Find the maximum profit P.
Full step-by-step solutions: practise this topic on examproblems.com.