Modelling with functionsAQA A-Level Maths: Subtopic test

1. 1
   The height, $h$ metres, of a ball above the ground $t$ seconds after it is thrown upwards is modelled by $h=1.5+12t-5t^2$ for $t\geq0$.

   (a)

   What does the constant $1.5$ represent in this model?

   [1 mark]

   • AThe greatest height reached by the ball
   • BThe height from which the ball is released
   • CThe speed with which the ball is thrown
   • DThe time at which the ball lands

   (b)

   Which of the following is a limitation of the model?

   [1 mark]

   • AIt predicts that the ball never reaches a greatest height
   • BIt assumes that the ball is released from ground level
   • CIt gives a negative height when $t=1$
   • DIt predicts negative heights for large $t$, which cannot happen after landing

   (c)

   Find the maximum height of the ball predicted by the model.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The number of bacteria, $N$, in a dish $t$ hours after the start of an experiment is modelled by $N=200\times1.5^t$.

   (a)

   How many bacteria does the model predict at the start of the experiment?

   [1 mark]

   • A$200$
   • B$300$
   • C$1.5$
   • D$0$

   (b)

   How many bacteria does the model predict after $2$ hours?

   [1 mark]

   • A$600$
   • B$400$
   • C$450$
   • D$300$

   (c)

   Explain why the model is unlikely to be valid for large values of $t$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A company models the monthly cost, $C$ pounds, of running a delivery van that travels $x$ miles in a month by $C=450+0.2x$.

   (a)

   Interpret the numbers $450$ and $0.2$ in the context of the model, and state one limitation of the model.

   [3 marks]

   (b)

   A second van is modelled by $D=300+0.35x$. Find the number of miles for which the two models give the same cost, and state, with working, which van is cheaper for a month in which $700$ miles are travelled.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The temperature, $\theta\,^\circ\text{C}$, of a cup of coffee $t$ minutes after it is poured is modelled by $\theta=20+70\times0.9^t$.

   (a)

   (i) Write down the initial temperature of the coffee.
   (ii) Find the temperature after $10$ minutes, to $3$ significant figures.
   (iii) Find the time taken for the coffee to cool to $50\,^\circ\text{C}$, giving your answer in minutes to $3$ significant figures.

   [6 marks]

   (b)

   (i) State the temperature that the model predicts the coffee approaches, and give a reason why this is sensible.
   (ii) The room is in fact at $18\,^\circ\text{C}$. Suggest a refined model of the form $\theta=18+A\times0.9^t$ that still gives $90\,^\circ\text{C}$ at $t=0$.
   (iii) Use the refined model to find the temperature after $10$ minutes, and compare it with your answer to part (a)(ii).

   [6 marks]

   Total for question 4: 12 marks

End of questions