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

A mobile plan charges a monthly fee plus a price for each GB of data. Using 4 GB costs £17 and using 10 GB costs £35.

1. (a) Find a model of the form \(C=md+c\).
2. (b) Find the cost of using 7.5 GB.

The weekly profit £P of a concert is modelled by \(P=-3x^2+84x-360\), where £x is the ticket price.

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

An open box is made from a rectangular card 36 cm by 24 cm. A square of side x cm is cut from each corner and the sides are folded up. The volume is \(V=x(36-2x)(24-2x)\) cm³.

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

The braking distance s metres of a car is directly proportional to the square of its speed v m/s: \(s=av^2\). At 20 m/s the braking distance is 24 m.

1. (a) Find a.
2. (b) Find the braking distance at 30 m/s.

A model is \(f(x)=12+30(0.8)^x\), for \(x\ge0\).

1. (a) Write down the equation of the horizontal asymptote.
2. (b) Find f(0).
3. (c) Solve f(x) = 20.

A tank holds 40 litres of water at t = 0, where t is in minutes, and 100 litres at t = 6. Water flows in at a constant rate. The tank holds at most 300 litres.

1. (a) Find a model \(V=mt+c\).
2. (b) Find the time when there are 250 litres.
3. (c) Write down a reasonable domain for the model.

A footbridge arch is modelled by \(h=-0.04x^2+1.6x\), where h metres is the height above the water and x metres is the horizontal distance from the left foot of the arch.

1. (a) Find the width of the arch at water level.
2. (b) Find the greatest height of the arch.
3. (c) Find the width of the arch at a height of 12 m.

The charge level C% of a phone battery t hours after it starts charging is modelled by \(C=100-85a^t\), with 0 < a < 1. After 2 hours the charge is 55%.

1. (a) Find a.
2. (b) Find the charge level after 5 hours.
3. (c) Find the time for the charge level to reach 95%.

The time T hours to paint a hall is inversely proportional to the number n of painters: \(T=kn^{-1}\). Three painters take 14 hours.

1. (a) Find k.
2. (b) Find the time taken by 7 painters.
3. (c) Find the least whole number of painters who can finish in under 4 hours.

The number N of people (in hundreds) at a festival t hours after it opens is modelled by \(N=-0.5t^3+6t^2\), for \(0\le t\le12\).

1. (a) Find when the attendance is greatest, and the greatest attendance.
2. (b) Find the length of time for which the attendance is at least 10 000.
3. (c) Explain why the model is only used for \(0\le t\le12\).

A cup of tea in a room at 25 °C is measured every 5 minutes. T °C is the temperature after t minutes. The data is: (0, 95), (5, 61), (10, 43), (15, 34), (20, 29). Two models are suggested. A linear model through (0, 95) and (20, 29), and the exponential model \(T=25+70(0.875)^t\).

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.