Problem solving and modellingAQA A-Level Maths: Topic test

1. 1
   A stone is dropped from rest from a bridge that is 50 m above the surface of a river. The stone is modelled as a particle moving under gravity only, with acceleration $9.8$ m s$^{-2}$ downwards.

   (a)

   Which of the following is a modelling assumption made when the stone is treated as a particle moving under gravity only?

   [1 mark]

   • AAir resistance on the stone is negligible
   • BThe stone has zero mass
   • CThe acceleration of the stone increases as it falls
   • DThe speed of the stone is constant

   (b)

   Using the model, what is the speed of the stone 3 seconds after it is dropped?

   [1 mark]

   • A$14.7$ m s$^{-1}$
   • B$9.8$ m s$^{-1}$
   • C$88.2$ m s$^{-1}$
   • D$29.4$ m s$^{-1}$

   (c)

   The stone is observed to take 3.4 s to reach the river. Use the model to find the time predicted for the stone to fall 50 m, and comment on the model in the light of the observed time.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A café owner wants to decide how many cups of coffee to prepare each morning. She works through the mathematical problem-solving cycle: specify the problem, collect information, process and represent the information, and interpret the results.

   (a)

   She records the number of cups sold on each of 20 mornings. At which stage of the cycle is she working?

   [1 mark]

   • ASpecify the problem
   • BCollect information
   • CProcess and represent the information
   • DInterpret the results

   (b)

   After calculating a mean of 85 cups and a standard deviation of 12 cups, she notices that Friday sales are much higher than on other days. What should she do next?

   [1 mark]

   • AStop, because the calculations are complete
   • BDelete the Friday figures so that the data is consistent
   • CRepeat the cycle, treating Fridays separately and refining her approach
   • DUse the mean of 85 cups for every day regardless

   (c)

   From the 20 mornings, the mean number of cups sold is 85 and the standard deviation is 12. She prepares 97 cups each morning, the mean plus one standard deviation. Give one reason, in context, why this is a sensible choice and one limitation of her data.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A kettle containing water at 20 $^{\circ}$C is switched on. The temperature $T$ $^{\circ}$C of the water $t$ seconds after it is switched on is modelled by $T=20+0.4t$.

   (a)

   Find the time predicted by the model for the water to reach 100 $^{\circ}$C, and state one assumption that the model makes about how the water is heated.

   [3 marks]

   (b)

   The temperature is observed to be 62 $^{\circ}$C after 120 seconds. Refine the model by changing the constant 0.4 to a value $k$ so that $T=20+kt$ fits this observation, and hence find the time at which the refined model predicts that the water reaches 100 $^{\circ}$C.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A ladder of length 4 m leans against a vertical wall with its foot on horizontal ground. It is modelled as a straight line. Safety guidance says that the angle $\theta$ between the ladder and the ground should be between $65^{\circ}$ and $75^{\circ}$ inclusive.

   (a)

   Construct a model for the distance $x$ metres of the foot of the ladder from the wall, and use it to find the range of values of $x$ that satisfy the guidance. Give your answers to 3 significant figures, and state one assumption of the model.

   [6 marks]

   (b)

   The top of the ladder must reach at least 3.7 m above the ground. Using your answer to part (a), find the range of values of $x$ for which the ladder both meets the safety guidance and reaches this height, and comment on one limitation of the model.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A student wants to estimate how many tennis balls will fit inside a classroom measuring 8 m by 6 m by 3 m. A tennis ball is modelled as a sphere of diameter 6.5 cm.

   (a)

   Ignoring gaps between the balls, which is the best estimate of the number of balls that fill the room?

   [1 mark]

   • A$10^{4}$
   • B$10^{5}$
   • C$10^{6}$
   • D$10^{8}$

   (b)

   Which statement best evaluates the estimate in part (a)?

   [1 mark]

   • AThe true number is larger, because the balls can be squashed together
   • BThe true number is smaller, because spheres cannot fill all the space and leave gaps
   • CThe estimate is exact, because the volumes are known
   • DThe true number is larger, because the room contains air

   (c)

   Spheres packed together fill at most about 74% of the available space. Use this to give a refined estimate of the number of tennis balls, in standard form to 2 significant figures.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A town had a population of 24 000 in 2020 and 25 200 in 2022. A planner models the population as increasing by the same number of people each year.

   (a)

   What population does the planner's model predict for 2030?

   [1 mark]

   • A27 600
   • B28 800
   • C36 000
   • D30 000

   (b)

   Which is a limitation of using this model to predict the population in 2060?

   [1 mark]

   • AIt assumes a steady increase in numbers, which may be unrealistic over a long period
   • BIt cannot be used because only two data points are known
   • CIt always predicts a negative population
   • DLinear models are never used for populations

   (c)

   A second planner models the population as $P=24\,000\times1.025^{t}$, where $t$ is the number of years after 2020. Find the population predicted for 2030 by this model, and state which planner's model predicts the higher value.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A school is organising a trip for 140 students and 9 staff, so 149 passengers in total, and every passenger needs a seat. Coaches seat 49 passengers and cost £420 each. Minibuses seat 16 passengers and cost £150 each.

   (a)

   Find the number of coaches needed, and the total cost, if only coaches are used.

   [3 marks]

   (b)

   The school may use any combination of coaches and minibuses. Find the cheapest combination that seats all 149 passengers, showing that your answer is cheaper than other combinations.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A theatre sells 400 stalls tickets for each performance when the ticket price is £20. A manager models that every £1 increase in the price, $p$ pounds, reduces the number of tickets sold by 8.

   (a)

   Construct a model for the number of tickets sold, $N$, and the revenue, $R$ pounds, from stalls tickets. Hence find the price that gives the maximum revenue, and the maximum revenue.

   [6 marks]

   (b)

   The stalls has only 300 seats. Using your model from part (a), find the range of prices for which the number of tickets sold is between 0 and 300 inclusive, and evaluate whether the maximum revenue in part (a) is still valid. State one further limitation of the model.

   [6 marks]

   Total for question 8: 12 marks

End of questions