Exponential models, half-life and logarithmic models Problem set B

The charge on a capacitor is \(Q=Q_0e^{-0.62t}\), with t in seconds. Find the time for the charge to halve.

Starting from \(P=P_0e^{kt}\), where k > 0, show that the doubling time is \(\dfrac{\ln2}{k}\).

A soil sample holds 120 units of salt where the background level is 45 units. The excess over the background halves every 5 days. Write the amount as \(S=d+Ae^{kt}\), with t in days.

A model \(y=a+b\ln x\) passes through (3, 2.5) and (30, 8). Find a and b.

A model is \(y=7.9-1.6\ln x\).

1. (a) Find y when x = 9.
2. (b) Find x when y = 3.

A tank of water is heated to 70°C in a cellar at 15°C. The temperature above the cellar halves every 45 minutes, where t is in minutes.

1. (a) Find a model \(T=d+Ae^{kt}\) for the temperature.
2. (b) Find when the temperature falls to 25°C.

A sample of iodine-131 has a mass of 250 g at t = 0, where t is in days. Its mass halves every 8 days.

1. (a) Write a model \(M=Ae^{kt}\) for the mass.
2. (b) Find the mass left after 20 days.

The temperature T°C of a lake at depth d metres is modelled by \(T=a+b\ln d\). It is 18.5°C at 3 m and 9°C at 25 m.

1. (a) Find a and b.
2. (b) Find the temperature at 10 m deep.
3. (c) Find the depth at which the temperature is 12°C.

The population of a town increases by 2.4% each year. It is modelled by \(P=P_0e^{kt}\), where t is in years.

1. (a) Find k.
2. (b) Find the time for the population to multiply by 2.5.

A fossil shell has 28% of the carbon-14 of a living shell. The half-life of carbon-14 is 5730 years. The amount is \(M=M_0e^{kt}\), where t is in years.

1. (a) Find k.
2. (b) Find the age of the shell.

The graph shows the depth D cm of water in a draining pool t hours after the plug is pulled. It is modelled by \(D=d+Ae^{kt}\).

1. (a) Write down d and A.
2. (b) Find k.
3. (c) Find D when t = 15.

A model is \(y=a+b\ln x\).

1. (a) Show that trebling x adds \(b\ln3\) to y, whatever x is.
2. (b) Find the increase in y when x trebles and b = 2.4.