Logarithmic graphs and modelling growth and decayEdexcel A-Level Maths: Subtopic test

1. 1
   Experimental data for two variables $x$ and $y$ are modelled by $y=ax^{n}$, where $a$ and $n$ are constants. When $\log_{10}y$ is plotted against $\log_{10}x$, the points lie on a straight line with gradient $2.5$ and vertical-axis intercept $0.6$.

   (a)

   What is the value of $n$?

   [1 mark]

   • A$0.6$
   • B$316$
   • C$2.5$
   • D$0.4$

   (b)

   What is the value of $a$, to 3 significant figures?

   [1 mark]

   • A$3.98$
   • B$0.6$
   • C$1.82$
   • D$0.251$

   (c)

   Use the model to find the value of $y$ when $x=4$, giving your answer to 3 significant figures.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Experimental data for two variables $t$ and $P$ are modelled by $P=kb^{t}$, where $k$ and $b$ are constants. When $\log_{10}P$ is plotted against $t$, the points lie on a straight line with gradient $0.30$ and vertical-axis intercept $1.20$.

   (a)

   Which statement about the line is correct?

   [1 mark]

   • AThe gradient is $b$ and the intercept is $k$.
   • BThe gradient is $\log_{10}k$ and the intercept is $\log_{10}b$.
   • CThe gradient is $k$ and the intercept is $\log_{10}b$.
   • DThe gradient is $\log_{10}b$ and the intercept is $\log_{10}k$.

   (b)

   What is the value of $b$, to 3 significant figures?

   [1 mark]

   • A$0.30$
   • B$2.00$
   • C$1.35$
   • D$15.8$

   (c)

   Find the value of $k$, to 3 significant figures, and hence write down the model for $P$ in terms of $t$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The concentration, $C$ mg per litre, of a drug in a patient's blood $t$ hours after an injection is modelled by $C=40e^{-0.25t}$ for $t\geq0$.

   (a)

   (i) State the concentration immediately after the injection.
   (ii) Find the concentration 6 hours after the injection, to 3 significant figures.

   [3 marks]

   (b)

   Find the time after the injection at which the concentration first falls to $10$ mg per litre. Give your answer in hours to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The population, $P$ thousand, of a species of bird on an island $t$ years after 1 January 2020 is modelled by $P=12e^{0.08t}$. The island's habitat can support at most $60$ thousand of these birds.

   (a)

   (i) State the population on 1 January 2020.
   (ii) Find the value of $t$ at which the model predicts a population of $30$ thousand, to 3 significant figures.
   (iii) Show that a graph of $\ln P$ against $t$ would be a straight line, and state its gradient and its vertical-axis intercept.

   [6 marks]

   (b)

   (i) Find the population that the model predicts when $t=100$.
   (ii) Find the value of $t$ at which the model first predicts a population greater than the habitat can support, to 3 significant figures.
   (iii) Comment on the validity of the model for values of $t$ after this time, and suggest one refinement.

   [6 marks]

   Total for question 4: 12 marks

End of questions