Mixed modelling Problem set B

Give each answer to 3 significant figures.

1. (a) A medical tracer has a half-life of 14 days. The mass is modelled by \(M = M_0e^{-kt}\), with t in days. Find k.
2. (b) The population of a town doubles every 25 years (t in years). The number is modelled by \(N = N_0e^{kt}\). Find k.
3. (c) The size of a sample is \(S = 900 \times 1.8^{-t}\), with t in years. Find its half-life.

A bottle of milk is taken from a fridge at 4 °C and warms up in a kitchen. Its temperature \(T\) °C, \(t\) minutes later, is \(T = B + Ae^{-kt}\). After 25 minutes it is 15 °C. The kitchen is at 23 °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 20 °C.

The mass of a radioactive waste sample is \(M = M_0e^{-0.045t}\) g, where t is the time in days since it was stored. After 14 days 80 g remain.

1. (a) Find the initial amount \(M_0\).
2. (b) Find the time at which the amount reaches 25 g.

The number of bacteria on a worktop after cleaning is modelled by \(N = 5000e^{-0.9t}\), where t is the time in hours.

1. (a) Use the model to find the number of bacteria after 8 hours.
2. (b) State one assumption of the model.
3. (c) Explain why the model cannot be right for large values of t.

A fitness score R is modelled by \(R = p\ln A + q\), where A is the number of hours of training in a month. When \(A = 2\), \(R = 35\). When \(A = 60\), \(R = 71\).

1. (a) Find the values of p and q.
2. (b) Find R when A = 25.
3. (c) Find the range of values of A for which \(50 \le R \le 65\).

A tank on a farm is filled by a pump. The volume \(V\) litres after \(t\) minutes is modelled by \(V = pt\) for \(0 \le t \le 5\) and \(V = qt^2\) for \(t > 5\). After 1.5 minutes there are 22.5 litres. The tank holds 900 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.

There are 900 frogs of a rare species in a nature reserve at \(t=0\), with t in years. The population halves every 6 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 pie leaves an oven at 190 °C into a kitchen at 22 °C. Its temperature is \(T = 22 + 168e^{-kt}\) °C, where t is in minutes. After 15 minutes, the difference between the pie and the kitchen has fallen to a fraction q of its starting value, where \(0 < q < 1\).

1. (a) Show that \(k = -\frac{1}{15}\ln q\).
2. (b) Find an expression for the time to reach 60 °C in terms of q.
3. (c) Find the time when \(q = 0.5\).

The temperature in a greenhouse is modelled by \(P\) °C, where t is the time in hours after midnight. The lowest temperature in a day is 2.5 °C and the highest is 6.5 °C. At midnight, \(P = 3\) and the temperature is rising. 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 greenhouse is modelled by Q, with the same period and middle value as P, half the amplitude, and its greatest value 4 hours later than P's. Write down Q in the same form.

Use P and Q from the last question (temperatures in two greenhouses, in °C). 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 number of users of a new app (in thousands) is modelled by \(P = \frac{800}{1 + Ce^{-kt}}\), where t is the time in months after the launch.

1. (a) Given that \(P = 640\) when \(t = 3\), show that \(C = \frac{1}{4}e^{3k}\).
2. (b) The value of P when \(t = 7\) is 760. 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 months for P to exceed 790.

A school's online club had 800 members at the start of the year. Let t be the number of months after that.

1. (a) A linear model assumes the number of members grows by 90 every month. Find the predicted number after 8 months.
2. (b) Give one reason why a linear model is unlikely to be suitable for a long time.
3. (c) A logistic model \(P = \frac{3000}{1+Ce^{-kt}}\) is used instead. Find C.
4. (d) After 5 months the number is 1900. Find k, to 1 decimal place.
5. (e) Use your value of k to predict the number after 10 months. Sketch the graph of P against t.

A sample of cells has 150 cells at \(t=0\), where t is in hours. The number doubles every 5 hours for \(0 \le t \le 20\), then halves every 6 hours for \(t > 20\).

1. (a) Find the number of cells at \(t = 20\).
2. (b) Find the number of cells at \(t = 30\).
3. (c) Find the time when the number of cells is back to 150.
4. (d) The doubling time is \(\alpha\) hours and the halving time is \(\beta\) hours. Find the condition on \(\alpha\) and \(\beta\) for the number of cells at \(t = 30\) to be greater than the starting number of cells.