Modelling with functionsEdexcel A-Level Maths: Subtopic test

1. 1
   The temperature, $T\,^\circ\text{C}$, of a cup of tea $t$ minutes after it is made is modelled by $T=20+70e^{-0.05t}$, $t\geq0$.

   (a)

   What does the model predict for the initial temperature of the tea?

   [1 mark]

   • A$70\,^\circ\text{C}$
   • B$20\,^\circ\text{C}$
   • C$50\,^\circ\text{C}$
   • D$90\,^\circ\text{C}$

   (b)

   What temperature does the model predict the tea approaches as $t$ becomes very large?

   [1 mark]

   • A$0\,^\circ\text{C}$
   • B$20\,^\circ\text{C}$
   • C$90\,^\circ\text{C}$
   • D$70\,^\circ\text{C}$

   (c)

   Find the time taken for the tea to cool to $50\,^\circ\text{C}$, giving your answer to $3$ significant figures.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The depth of water, $d$ metres, in a harbour $t$ hours after midnight is modelled by $d=6+2.5\sin(30t)^\circ$, $0\leq t\leq24$.

   (a)

   What is the period of the depth model, in hours?

   [1 mark]

   • A$6$
   • B$30$
   • C$12$
   • D$24$

   (b)

   What is the minimum depth of water predicted by the model?

   [1 mark]

   • A$3.5$ m
   • B$2.5$ m
   • C$6$ m
   • D$8.5$ m

   (c)

   Find the first time after midnight at which the depth is $4.75$ m, giving your answer as a time of day.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The pressure, $P$ kPa, of a fixed mass of gas at constant temperature is modelled as inversely proportional to its volume, $V\,\text{cm}^3$. When $V=250$, $P=120$.

   (a)

   Show that $P=\frac{30000}{V}$ and use the model to find the pressure when $V=80$.

   [3 marks]

   (b)

   A student compresses the gas until $V=20$ and measures $P=1200$ kPa.
   (i) Find the percentage by which the model's prediction exceeds the measured pressure.
   (ii) Suggest a reason for the difference, and a way to refine the model.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The number of bacteria, $N$, in a culture $t$ hours after the start of an experiment is modelled by $N=500e^{0.4t}$, $t\geq0$.

   (a)

   (i) State the initial number of bacteria.
   (ii) Find the time taken for the number of bacteria to reach $10000$, giving your answer to $3$ significant figures.
   (iii) Find the rate of increase of the number of bacteria at $t=5$, giving your answer to $3$ significant figures.

   [6 marks]

   (b)

   A scientist suggests the refined model $N=\frac{20000}{1+39e^{-0.4t}}$.
   (i) Show that the refined model gives the same initial number of bacteria.
   (ii) State the number of bacteria the refined model predicts in the long term.
   (iii) Find the number of bacteria predicted by each model after $10$ hours, giving your answers to $3$ significant figures, and explain which model is more realistic for large $t$.

   [6 marks]

   Total for question 4: 12 marks

End of questions