Optimisation in context Problem set A

The power output of a solar panel, P watts, at t hours after midnight is modelled by \(P=-0.8t^2+19.2t-90\), for \(7\le t\le 17\). Find

1. (a) the time when the output is greatest;
2. (b) the greatest output.

The running cost of a lorry is \(C=0.01v^2+\frac{160}{v}\) euros per kilometre, where v km/h is its speed. Find

1. (a) the speed that makes the cost smallest;
2. (b) the smallest cost per kilometre.

A festival sells a fraction s of its tickets in advance. Its revenue, in thousands of euros, is modelled by \(R=30s(1-s)^2\), for \(0\le s\le 1\). Find

1. (a) the fraction that gives the greatest revenue;
2. (b) the greatest revenue.

A closed box has a rectangular base measuring \(2x\) cm by x cm, and a height of h cm. Its total surface area is 540 cm².

1. (a) Write the height h in terms of x.
2. (b) Write the volume V in terms of x only.
3. (c) Find the greatest possible volume, and the value of x that gives it.

A farmer has 160 m of fencing to make a rectangular pen against a straight wall. The wall forms one side, so only three sides need fencing. The two sides at right angles to the wall are x metres long.

1. (a) Write the area A in terms of x.
2. (b) Find the greatest possible area, and the value of x that gives it.

A rectangular sheet of card measures 60 cm by 36 cm. A square of side x cm is cut from each corner and the sides are folded up to make an open gift box.

1. (a) Write the length and the width of the base in terms of x.
2. (b) Write the volume V in terms of x, and state the values x can take.
3. (c) Find the greatest possible volume, and the value of x that gives it.

The flow rate of a river, F cubic metres per second, at a distance d hundred metres downstream of a weir is modelled by \(F=-0.01d^3+0.3d^2\), for \(0\le d\le 30\). Find

1. (a) the distance where the flow is greatest;
2. (b) the greatest flow rate.

A storage tank has no lid. It has a square base of side x metres and a height of h metres, and its volume is 32 m³. The tank is made from steel sheet.

1. (a) Write the height h in terms of x.
2. (b) Write the surface area S in terms of x only.
3. (c) Find the smallest possible surface area, and the value of x that gives it.

A bakery sells 250 loaves a day at €3 each. For every 10 cent rise in the price, 6 fewer loaves are sold each day. Each loaf costs €1 to make. Let x be the number of 10 cent rises.

1. (a) Write an expression for the price and for the number sold after x rises.
2. (b) Write an expression for the daily profit P in euros.
3. (c) Find the greatest profit when x must be a whole number, and the price that gives it.

A school running track has two straight sides of length L metres. At each end there is a semicircle of diameter w metres. The perimeter of the track is 300 m.

1. (a) Write L in terms of w.
2. (b) Write the area A of the rectangle in terms of w.
3. (c) Find the greatest possible area of the rectangle, and the value of w that gives it.

A closed cylindrical tin has radius r cm and height h cm. Its total surface area is 600 cm².

1. (a) Write V in terms of r only.
2. (b) Find the greatest possible volume, and the value of r that gives it.

A rectangular plot measuring x metres by y metres has an area of 800 m². It is fenced. Fencing along the two sides of length x costs €15 per metre, and along the two sides of length y costs €6 per metre.

1. (a) Write the cost C in terms of x only.
2. (b) Find the least possible cost, and the dimensions of the rectangle.