Optimisation in context Problem set B

The number of customers per hour entering a café, S, at t hours after midnight is modelled by \(S=-1.5t^2+33t-120\), for \(6\le t\le 16\). Find

1. (a) the time when the number is greatest;
2. (b) the greatest number of customers per hour.

The fuel cost of a ship is \(C=0.5v^2+\frac{1728}{v}\) dollars per hour, where v knots is its speed. Find

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

A bakery makes a fraction f of its dough into rolls. Its profit, in hundreds of pounds, is modelled by \(P=12f(1-f)^2\), for \(0\le f\le 1\). Find

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

A closed storage crate has a rectangular base measuring \(3x\) cm by x cm, and a height of h cm. Its total surface area is 864 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 gardener has 84 m of edging to enclose a rectangular flower bed against a straight garden wall. The wall forms one side, so only three sides need edging. 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 tin measures 54 cm by 30 cm. A square of side x cm is cut from each corner and the sides are folded up to make an open planter.

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 speed of a rocket sled, v metres per second, at t seconds after the start is modelled by \(v=-0.02t^3+0.45t^2\), for \(0\le t\le 22.5\). Find

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

A fish crate has no lid. It has a square base of side x cm and a height of h cm, and its volume is 4000 cm³. The crate is made from plastic 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 cinema sells 400 tickets for a film at €10 each. For every 50 cent rise in the price, 15 fewer tickets are sold. Each ticket costs the cinema €2. Let x be the number of 50 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 profit P in euros.
3. (c) Find the greatest profit when x must be a whole number, and the price that gives it.

A swimming pool is a rectangle of length L metres with a semicircle of diameter w metres at each end. The distance all the way round the edge of the pool is 160 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 pencil case has radius r cm and height h cm. Its total surface area is 900 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 patio measuring x metres by y metres has an area of 450 m². It is edged with stone. The edging along the two sides of length x costs £10 per metre, and along the two sides of length y costs £4 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.