Mathematical models: exam review Exam review

The amount M mg of a medicine in a patient's blood, t hours after it is taken, is modelled by \(M=M_0e^{-kt}\). [7]

1. (a) The half-life of the medicine is 6 hours. Find the value of k. Give your answer to 3 significant figures. [2]
2. (b) After 10 hours the amount is 48 mg. Find \(M_0\). Use the unrounded value of k. [2]
3. (c) Find the time for the amount to fall to 10 mg. Give your answer to 3 significant figures. [2]
4. (d) State one assumption made by this model. [1]

A learner's typing speed S (words per minute) after t weeks of practice is modelled by \(S=a+b\ln t\). After 2 weeks the speed is 35. After 10 weeks the speed is 62. [6]

1. (a) Use the information to find a and b. Give each value to 3 significant figures. [3]
2. (b) Use the model to predict S when t = 20. [1]
3. (c) Find the value of t when S = 80. [2]

A cup of tea is made at 92°C in a room at 21°C. Its temperature T°C, t minutes later, is modelled by \(T=B+Ae^{-kt}\). [7]

1. (a) Write down the value of B and the value of A. [2]
2. (b) After 5 minutes the temperature is 70°C. Find the value of k. Give your answer to 3 significant figures. [2]
3. (c) Find the time for the temperature to reach 40°C. Give your answer to 3 significant figures. [2]
4. (d) Find the half-life of the difference between the temperature and the surroundings. Give your answer to 3 significant figures. [1]

A tank is filled with water. The volume V litres after t minutes is modelled by \(V=pt\) for \(0\le t\le 6\), and \(V=qt^2\) for \(t>6\). After 3 minutes the volume is 120 litres. [5]

1. (a) Find the value of p. [1]
2. (b) The model is continuous at t = 6. Find the value of q. [2]
3. (c) Find the time at which the volume reaches 1000 litres. Give your answer to 3 significant figures. [2]

In each part, the function is continuous at the point where the rule changes. [8]

1. (a) \(f(x)=\begin{cases}3x^2+a & x\le 2\\ 5x-1 & x>2\end{cases}\). Find the value of a. [2]
2. (b) \(g(x)=\begin{cases}x^a & x\le 9\\ 2x+9 & x>9\end{cases}\). Find the value of a. Give your answer to 3 significant figures. [3]
3. (c) \(h(x)=\begin{cases}e^{ax} & x\le 2\\ 4x+2 & x>2\end{cases}\). Find the value of a. Give your answer to 3 significant figures. [3]

In each part, give answers that are not exact to 3 significant figures. [9]

1. (a) The time t hours taken to empty a tank is inversely proportional to the number n of pumps used, so \(t=\dfrac{a}{n}\). With 4 pumps the tank is emptied in 9 hours. Find the value of a, and the time when n = 6. [3]
2. (b) Describe what the model predicts for the time as n gets close to 0, and say why this is not reasonable. [1]
3. (c) A drink is put into a fridge that is kept at 4 °C. Its temperature T °C after t minutes is modelled by \(T=c+k(0.92)^t\), where c and k are constants. At t = 0 the temperature is 25 °C. Write down the equation of the horizontal asymptote of the graph of T, and find the value of k. [3]
4. (d) Find the time for the temperature to reach 8 °C. Give your answer to 3 significant figures. [2]

In each part, the amount is modelled by \(y=y_0e^{kt}\), where t is in years. [5]

1. (a) The value of a machine falls by 12% each year. Find the value of k. Give your answer to 3 significant figures. [2]
2. (b) A fossil has 35% of the carbon-14 that a living organism has. The half-life of carbon-14 is 5730 years. Find the age of the fossil. Give your answer to 3 significant figures. [3]

A tray is made from a rectangular sheet of metal 50 cm by 30 cm. A square of side x cm is cut from each corner and the sides are folded up. The volume of the tray is \(V=x(50-2x)(30-2x)\) cm³. [3]

1. (a) Write down the domain of x for which V makes sense. [1]
2. (b) Use your calculator to find the greatest volume, and the value of x that gives it. Give your answers to 3 significant figures. [2]

The height h metres of the sea at a harbour, t hours after midnight, can be modelled by a sinusoidal function. The greatest height is 5.8 m and the least height is 1.2 m. The height repeats every 12.4 hours. [12]

1. (a) Find the amplitude and the mean value of h. [2]
2. (b) The period is 12.4 hours. Find the value of b. Give your answer to 3 significant figures. [2]
3. (c) The graph of h first crosses its mean value, going upwards, at t = 2.1. Write down a model of the form \(h=a\sin(b(t-c))+d\). Use b to 3 significant figures. [1]
4. (d) Use your model to find h when t = 5. [2]
5. (e) Find the total time, in the first 12.4 hours, for which h is greater than 4. Give your answer to 3 significant figures. [3]
6. (f) Find the first time at which h is a maximum. [2]

The number P of fish in a lake, t years after fish are introduced, is modelled by \(P=\dfrac{L}{1+Ce^{-kt}}\). The lake can support at most 1200 fish, so L = 1200. Initially there are 150 fish. [12]

1. (a) Find the value of C. [2]
2. (b) After 3 years the number of fish is 420. Find the value of k. Give your answer to 3 significant figures. [3]
3. (c) Find the number of fish after 6 years. [2]
4. (d) Find the time at which the number of fish is half of 1200. Give your answer to 3 significant figures. [2]
5. (e) Write down the limiting value of the number of fish as t increases. [1]
6. (f) Give one reason why a logistic model is more realistic than an exponential model for this situation. [2]

A scientist studies two cultures. Culture X starts with 900 units and loses 7% of its amount each day. Culture Y starts with 40 units. [8]

1. (a) Write down an expression for the amount of culture X after n days. [1]
2. (b) Find the least whole number of days after which there is less than 100 units of culture X. [3]
3. (c) The amount of culture Y is modelled by \(Y=B_0e^{kt}\) with \(B_0=40\). It doubles every 12 days. Find the value of k. Give your answer to 3 significant figures. [2]
4. (d) Find the time for the amount of culture Y to reach 900 units. Give your answer to 3 significant figures. [2]

A theatre sells n tickets for a show when the ticket price is £p. When the price is £10, 200 tickets are sold. When the price is £16, 128 tickets are sold. The theatre uses a linear model for n in terms of p. [9]

1. (a) Find a model of the form \(n=mp+c\). [3]
2. (b) State what the value of m means in this context. [1]
3. (c) The income from ticket sales is \(R=p\,n\) pounds. Find the ticket price that gives the greatest income, and that income. Give your answers to 3 significant figures. [2]
4. (d) Find the range of ticket prices for which the income is more than £2000. Give your answers to 3 significant figures. [2]
5. (e) State a reasonable domain for p, and say why the model should not be used for a price of £30. [1]

A Ferris wheel has its lowest point 2 m above the ground and its highest point 32 m above the ground. It makes one complete turn every 8 minutes. A rider gets on at the lowest point at t = 0, where t is in minutes. The height of the rider above the ground is modelled by \(h=a\cos(bt)+d\) metres. [9]

1. (a) Find the values of a, b and d in the model \(h=a\cos(bt)+d\). Give b to 3 significant figures. [3]
2. (b) Find the height of the rider at t = 3. Give your answer to 3 significant figures. [1]
3. (c) Find the total time in one turn for which the height is more than 25 m. Give your answer to 3 significant figures. [3]
4. (d) State what the value of |a| represents in this context. [1]
5. (e) Give one reason why the model may not be reasonable. [1]