Mathematical modellingAQA A-Level Maths: Subtopic test

1. 1
   A ball is thrown upwards from a height of 1.5 m. Its height $h$ metres after $t$ seconds is modelled by $h=1.5+12t-5t^2$ for $t\ge0$.

   (a)

   Which modelling assumption is made in this model?

   [1 mark]

   • AThe ball experiences a large constant drag force
   • BAir resistance is negligible
   • CThe mass of the ball changes over time
   • DGravity varies significantly with height

   (b)

   According to the model, what is the maximum height reached by the ball?

   [1 mark]

   • A$8.7$ m
   • B$7.2$ m
   • C$13.5$ m
   • D$1.5$ m

   (c)

   The model is only valid for part of the ball's flight. Find the range of values of $t$ for which it is valid, giving your reasoning.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A population of bacteria is modelled by $P=200\times1.5^t$, where $P$ is the number of bacteria and $t$ is the time in hours after the start of an experiment.

   (a)

   Which assumption does the model make about the growth of the population?

   [1 mark]

   • AA constant number of bacteria is added each hour
   • BThe population shrinks over time
   • CThe population stays the same
   • DThe population is multiplied by a constant factor each hour, with no upper limit

   (b)

   According to the model, how many bacteria are there after 4 hours, to 3 significant figures?

   [1 mark]

   • A$1200$
   • B$800$
   • C$1010$
   • D$3200$

   (c)

   The model predicts about $665\,000$ bacteria after 20 hours, but in the experiment the population levels off at about 5000. Explain why the model fails, and suggest a refinement.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A van hire company's cost model is $C=40+25d$, where $C$ is the cost in pounds and $d$ is the number of days of hire.

   (a)

   Interpret the numbers $40$ and $25$ in the model, and state one assumption it makes.

   [3 marks]

   (b)

   The company changes its prices. The fixed charge of £40 and the rate of £25 per day still apply for the first 7 days, but each further day costs £15. (i) Write down a refined model for $C$ when $d>7$. (ii) Find the cost of a 10-day hire.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A website's daily number of visitors, $N$, on day $t$ after its launch is modelled by $N=-2t^2+60t+100$ for $0\le t\le30$.

   (a)

   (i) Find $N$ when $t=10$, and interpret the value of $N$ when $t=0$. (ii) By completing the square, find the day on which the number of visitors is greatest and state that number.

   [6 marks]

   (b)

   On day 40 the site actually received 400 visitors. (i) Find the value of $N$ that the model gives for $t=40$. (ii) Give two reasons why this shows that the model is limited. (iii) Find the largest whole number of days after launch for which the model gives a positive number of visitors. (iv) Suggest one way the model could be refined.

   [6 marks]

   Total for question 4: 12 marks

End of questions