Linear, quadratic, cubic, variation and exponential models Problem set B

A car-hire firm charges a fixed fee plus a price for each day. Hiring for 3 days costs £96 and hiring for 8 days costs £226.

1. (a) Find a model of the form \(C=md+c\).
2. (b) Find the cost of hiring for 6 days.

The daily profit £P of a bakery stall is modelled by \(P=-2x^2+52x-240\), where £x is the price of a loaf box.

1. (a) Find the price that gives the greatest profit, and that profit.
2. (b) Find the two prices at which the profit is zero.

A gardener makes an open planter from a rectangular sheet of plywood 50 cm by 30 cm, by cutting a square of side x cm from each corner and folding up the sides. The volume is \(V=x(50-2x)(30-2x)\) cm³.

1. (a) Write down a reasonable domain for x.
2. (b) Find the volume when x = 4.
3. (c) Find the greatest volume.

The power P kilowatts of a wind turbine is modelled by \(P=av^3\), where v m/s is the wind speed. At 6 m/s the power is 54 kW.

1. (a) Find a.
2. (b) Find the power at 10 m/s.

A model is \(f(x)=5+24(0.85)^x\), for \(x\ge0\).

1. (a) Write down the equation of the horizontal asymptote.
2. (b) Find f(0).
3. (c) Solve f(x) = 14.

A bath holds 18 litres of water at t = 0, where t is in minutes, and 66 litres at t = 4. Water flows in at a constant rate. The bath holds at most 150 litres.

1. (a) Find a model \(V=mt+c\).
2. (b) Find the time when there are 120 litres.
3. (c) Write down a reasonable domain for the model.

A tunnel entrance is modelled by \(h=-0.05x^2+1.5x\), where h metres is the height above the road and x metres is the horizontal distance from the left wall.

1. (a) Find the width of the tunnel at road level.
2. (b) Find the greatest height of the tunnel.
3. (c) Find the width of the tunnel at a height of 8 m.

A hot metal rod cools in a room. Its temperature T °C after t minutes is modelled by \(T=21+79a^t\), with 0 < a < 1. After 4 minutes the temperature is 50 °C.

1. (a) Find a.
2. (b) Find the temperature after 10 minutes.
3. (c) Find the time for the temperature to fall to 25 °C.

The time D days to dig a trench is inversely proportional to the number n of workers: \(D=kn^{-1}\). Five workers take 18 days.

1. (a) Find k.
2. (b) Find the time taken by 6 workers.
3. (c) Find the least whole number of workers who can finish in under 8 days.

The volume V (in millions of litres) of water in a reservoir t weeks after the rains stop is modelled by \(V=-t^3+9t^2\), for \(0\le t\le9\).

1. (a) Find when the volume is greatest, and the greatest volume.
2. (b) Find the length of time for which the volume is at least 80 million litres.
3. (c) Explain why the model is only used for \(0\le t\le9\).

A bowl of soup in a room at 18 °C is measured every 5 minutes. T °C is the temperature after t minutes. The data is: (0, 90), (5, 54), (10, 36), (15, 27), (20, 22). Two models are suggested. A linear model through (0, 90) and (20, 22), and the exponential model \(T=18+72(0.87)^t\).

1. (a) Find the linear model \(T=c+mt\).
2. (b) Find the temperature at t = 40 minutes predicted by the linear model, and say why it is unreasonable.
3. (c) Find the temperature at t = 40 minutes predicted by the exponential model.
4. (d) Which model is better for long times? Give a reason.