Exponential growth and decayAQA A-Level Maths: Subtopic test

1. 1
   The mass $M$ grams of a radioactive isotope, $t$ days after it was first measured, is modelled by $M=80\mathrm{e}^{-0.05t}$.

   (a)

   Find the mass of the isotope after 10 days.

   [1 mark]

   • A$79.5$ g
   • B$48.5$ g
   • C$76.1$ g
   • D$0.607$ g

   (b)

   Find the half-life of the isotope.

   [1 mark]

   • A$0.0721$ days
   • B$800$ days
   • C$10$ days
   • D$13.9$ days

   (c)

   Find the time taken for the mass of the isotope to fall to 10 g. Give your answer to 3 significant figures.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   £6000 is invested in an account with interest compounded continuously, so that after $t$ years the value of the investment is £$V$, where $V=6000\mathrm{e}^{0.04t}$.

   (a)

   Find the value of the investment after 5 years, to the nearest pound.

   [1 mark]

   • A£7200
   • B£7300
   • C£7328
   • D£6245

   (b)

   How long does it take for the investment to double in value?

   [1 mark]

   • A$17.3$ years
   • B$25$ years
   • C$34.7$ years
   • D$0.0577$ years

   (c)

   Find the equivalent annual rate of interest, as a percentage to 3 significant figures, that would give the same value after each full year.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A patient is given a dose of a drug. The concentration $C$ of the drug in the blood, in mg per litre, $t$ hours after the dose is modelled by $C=12\mathrm{e}^{-0.3t}$.

   (a)

   The drug is only effective while the concentration is at least 2 mg per litre. Find how long after the dose the drug stops being effective, giving your answer in hours to 3 significant figures.

   [3 marks]

   (b)

   Show that the time taken for the concentration to halve does not depend on when it is measured, and find this time in hours to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The population $P$ of deer on an island, $t$ years after counting began, is modelled by $P=250\mathrm{e}^{0.12t}$. The island's food supply could support at most 3000 deer.

   (a)

   (i) Find the population predicted by the model after 10 years.
   (ii) Find the time at which the model predicts the population first reaches 2000. Give your answer in years to 3 significant figures.
   (iii) Interpret the value $0.12$ in the model.

   [6 marks]

   (b)

   (i) Find the time at which the model predicts a population of 3000. Give your answer in years to 3 significant figures.
   (ii) Explain why the model is unlikely to be valid for large values of $t$.
   (iii) Suggest a refinement to the model.

   [6 marks]

   Total for question 4: 12 marks

End of questions