Exponential growth and decay modelsEdexcel International A Level Maths: Subtopic test

1. 1
   The population $P$ of a colony of bacteria is modelled by $P=400e^{0.05t}$, where $t$ is the time in hours after the start of observation.

   (a)

   What is the initial population of the colony?

   [1 mark]

   • A1
   • B$0.05$
   • C20
   • D400

   (b)

   Find the population when $t=10$, to the nearest whole number.

   [1 mark]

   • A652
   • B659
   • C200
   • D59365

   (c)

   Find the time taken for the population to double, giving your answer in hours to 3 significant figures.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The mass $m$ grams of a radioactive isotope is modelled by $m=50e^{-0.02t}$, where $t$ is the time in years after the isotope is first measured.

   (a)

   Find the rate of change of the mass, $\frac{dm}{dt}$, at $t=0$.

   [1 mark]

   • A$-1$ g per year
   • B$1$ g per year
   • C$-0.02$ g per year
   • D$-50$ g per year

   (b)

   What does the model predict about $m$ for large values of $t$?

   [1 mark]

   • A$m$ gets closer to $50$
   • B$m$ becomes negative
   • C$m$ gets closer and closer to $0$ but never reaches it
   • D$m$ reaches $0$ after $50$ years

   (c)

   Find the time taken for the mass to fall to 20 g, giving your answer in years to 3 significant figures.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A cup of tea is poured at time $t=0$ and left to cool in a room. Its temperature $\theta\,^\circ$C after $t$ minutes is modelled by $\theta=20+70e^{-0.04t}$.

   (a)

   Find the initial temperature of the tea, and the time taken for the tea to cool to $50^\circ$C, giving the time to 3 significant figures.

   [3 marks]

   (b)

   Find the rate at which the tea is cooling when $t=10$. Describe what the model predicts for the temperature after a very long time, and state whether this is realistic.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The population $P$ thousand of a town, $t$ years after the start of 2020, is modelled by $P=Ae^{kt}$, where $A$ and $k$ are constants. The population was 25 thousand at the start of 2020 and 31 thousand at the start of 2025. A calculator may be used.

   (a)

   Find the values of $A$ and $k$, and use the model to predict the population at the start of 2035, giving your answers to 3 significant figures.

   [6 marks]

   (b)

   A second model is proposed: $P=60-35e^{-0.04t}$. (i) Show that this model also gives 25 thousand at the start of 2020. (ii) Find the population this model predicts for the start of 2035 and compare it with your answer to (a). (iii) Explain why the second model is more suitable for predicting the population in the long term.

   [6 marks]

   Total for question 4: 12 marks

End of questions