Mixed modelling Problem set A

Give each answer to 3 significant figures.

1. (a) A radioactive isotope has a half-life of 6 days. The mass is modelled by \(M = M_0e^{-kt}\), with t in days. Find k.
2. (b) A population of yeast cells doubles every 40 minutes (t in minutes). The number is modelled by \(N = N_0e^{kt}\). Find k.
3. (c) The size of a sample is \(S = 500 \times 2.5^{-t}\), with t in years. Find its half-life.

A metal block cools in a workshop. Its temperature \(T\) °C, \(t\) minutes after it leaves a furnace, is \(T = B + Ae^{-kt}\). The block leaves the furnace at 140 °C. After 9 minutes it is 71 °C. The workshop is at 16 °C.

1. (a) Find the values of A and B. Say what each one means.
2. (b) Find the value of k.
3. (c) Find the time for the temperature to reach 30 °C.

The mass of a drug in a patient's blood is \(M = M_0e^{-0.18t}\) mg, where t is the time in hours after the dose. After 5 hours 12 mg remain.

1. (a) Find the initial amount \(M_0\).
2. (b) Find the time at which the amount reaches 2 mg.

The number of fish in a lake is modelled by \(N = 2400e^{-0.35t}\), where t is the time in years after a spill.

1. (a) Use the model to find the number of fish after 20 years.
2. (b) State one assumption of the model.
3. (c) Explain why the model cannot be right for large values of t.

The rating R of an earthquake is modelled by \(R = p\ln A + q\), where A is the amplitude of the ground movement in micrometres. When \(A = 20\), \(R = 4.1\). When \(A = 500\), \(R = 6.3\).

1. (a) Find the values of p and q.
2. (b) Find R when A = 8000.
3. (c) Find the range of values of A for which \(5 \le R \le 6\).

A paddling pool is filled. 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 there are 36 litres. The pool holds 700 litres.

1. (a) Find the value of p.
2. (b) The model is continuous at the switch. Find the value of q.
3. (c) Find the time taken to fill the container.

A rare plant has 6000 plants at \(t=0\), where t is in years. The number is halving every 4 years.

1. (a) Write down a model \(N = N_0e^{-kt}\), giving N0 and k.
2. (b) Use the model to find when the number first falls below 1.
3. (c) Explain why the model cannot tell you exactly when the species dies out.

A cup of coffee cools in a room at 20 °C. Its temperature is \(T = 20 + 70e^{-kt}\) °C, where t is in minutes. After 10 minutes, the difference between the coffee and the room has fallen to a fraction q of its starting value, where \(0 < q < 1\).

1. (a) Show that \(k = -\frac{1}{10}\ln q\).
2. (b) Find an expression for the time to reach 35 °C in terms of q.
3. (c) Find the time when \(q = 0.4\).

The power used in a town is modelled by \(P\) megawatts, where t is the time in hours after midnight. The least power used in a day is 5 MW and the greatest is 9 MW. At midnight, \(P = 6\) and the power is increasing. The pattern repeats every day.

1. (a) Find a model \(P = a\sin(b(t-c)) + d\) with \(a > 0\) and \(0 \le c < 24\). Give c to 3 significant figures.
2. (b) A second model, Q, has the same period and the same middle value as P, half the amplitude, and its greatest value comes 3 hours later than P's. Write down Q in the same form.

Use P and Q from the last question (power used by two towns, in MW). Use technology.

1. (a) Find the first time after \(t=0\) when \(P = Q\).
2. (b) Find the maximum value of \(P + Q\) and the time it occurs, for \(0 \le t \le 24\).
3. (c) Sketch \(P+Q\), labelling its maximum and minimum points.
4. (d) Write \(P + Q\) in the form \(a\sin(b(t-c)) + d\).

The population of a species in a reserve is modelled by \(P = \frac{5000}{1 + Ce^{-kt}}\), where t is the time in years after it is introduced.

1. (a) Given that \(P = 3000\) when \(t = 2\), show that \(C = \frac{2}{3}e^{2k}\).
2. (b) The value of P when \(t = 6\) is 4500. Write down a second equation in C and k.
3. (c) Use technology to solve the two equations for k and C.
4. (d) Find the value of P when \(t = 0\).
5. (e) Find the least whole number of years for P to exceed 4800.

The population of a town was 2000 at the start of 2020. Let t be the number of years after the start of 2020.

1. (a) A linear model assumes the population grows by 150 every year. Find the predicted number after 12 years.
2. (b) Give one reason why a linear model is unlikely to be suitable for a long time.
3. (c) A logistic model \(P = \frac{5000}{1+Ce^{-kt}}\) is used instead. Find C.
4. (d) After 4 years the number is 3200. Find k, to 1 decimal place.
5. (e) Use your value of k to predict the number after 12 years. Sketch the graph of P against t.

A colony of insects has 300 insects at \(t=0\), where t is in days. The number doubles every 4 days for \(0 \le t \le 12\), then halves every 5 days for \(t > 12\).

1. (a) Find the number of insects at \(t = 12\).
2. (b) Find the number of insects at \(t = 30\).
3. (c) Find the time when the number of insects is back to 300.
4. (d) The doubling time is \(\alpha\) days and the halving time is \(\beta\) days. Find the condition on \(\alpha\) and \(\beta\) for the number of insects at \(t = 30\) to be greater than the starting number of insects.