Half-life and doubling time

3 \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-04_527_634_255_717} The number of bacteria in a population, \(P\), at time \(t\) hours is modelled by the equation \(P = a \mathrm { e } ^ { k t }\), where \(a\) and \(k\) are constants. The graph of \(\ln P\) against \(t\), shown in the diagram, has gradient \(\frac { 1 } { 20 }\) and intersects the vertical axis at \(( 0,3 )\).

1. State the value of \(k\) and find the value of \(a\) correct to 2 significant figures.
2. Find the time taken for \(P\) to double. Give your answer correct to the nearest hour. \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-05_2723_33_99_22}

5 A substance is decaying in such a way that its mass, m kg , at a time t years from now is given by the formula $$\mathrm { m } = 240 \mathrm { e } ^ { - 0.04 \mathrm { t } }$$

1. Find the time taken for the substance to halve its mass.
2. Find the value of \(t\) for which the mass is decreasing at a rate of 2.1 kg per year.

6 The mass \(M \mathrm {~kg}\) of a radioactive material is modelled by the equation $$M = M _ { 0 } \mathrm { e } ^ { - k t } ,$$ where \(M _ { 0 }\) is the initial mass, \(t\) is the time in years, and \(k\) is a constant which measures the rate of radioactive decay.

1. Sketch the graph of \(M\) against \(t\).
2. For Carbon \(14 , k = 0.000121\). Verify that after 5730 years the mass \(M\) has reduced to approximately half the initial mass. The half-life of a radioactive material is the time taken for its mass to reduce to exactly half the initial mass.
3. Show that, in general, the half-life \(T\) is given by \(T = \frac { \ln 2 } { k }\).
4. Hence find the half-life of Plutonium 239, given that for this material \(k = 2.88 \times 10 ^ { - 5 }\).

2 A radioactive substance decays exponentially, so that its mass \(M\) grams can be modelled by the equation \(M = A \mathrm { e } ^ { - k t }\), where \(t\) is the time in years, and \(A\) and \(k\) are positive constants.

1. An initial mass of 100 grams of the substance decays to 50 grams in 1500 years. Find \(A\) and \(k\).
2. The substance becomes safe when \(99 \%\) of its initial mass has decayed. Find how long it will take before the substance becomes safe.

5 A model for the radioactive decay of a form of iodine is given by $$m = m _ { 0 } 2 ^ { - \frac { 1 } { 8 } t }$$ The mass of the iodine after \(t\) days is \(m\) grams. Its initial mass is \(m _ { 0 }\) grams.

1. Use the given model to find the mass that remains after 10 grams of this form of iodine have decayed for 14 days, giving your answer to the nearest gram.
2. A mass of \(m _ { 0 }\) grams of this form of iodine decays to \(\frac { m _ { 0 } } { 16 }\) grams in \(d\) days. Find the value of \(d\).
3. After \(n\) days, a mass of this form of iodine has decayed to less than \(1 \%\) of its initial mass. Find the minimum integer value of \(n\).

The number of radioactive atoms, \(N\), in a sample of a sodium isotope after time \(t\) hours can be modelled by $$N = N_0 e^{-kt}$$ where \(N_0\) is the initial number of radioactive atoms in the sample and \(k\) is a positive constant. The model remains valid for large numbers of atoms.

1. It takes 15.9 hours for half of the sodium atoms to decay. Determine the number of days required for at least 90\% of the number of atoms in the original sample to decay. [5 marks]
2. Find the percentage of the atoms remaining after the first week. Give your answer to two significant figures. [2 marks]
3. Explain why the model can only provide an estimate for the number of remaining atoms. [1 mark]
4. Explain why the model is invalid in the long run. [1 mark]