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A driver is road-testing two minibuses, A and B, for a taxi company - AQA - A-Level Maths Pure - Question 15 - 2018 - Paper 2
Question 15
A driver is road-testing two minibuses, A and B, for a taxi company. The performance of each minibus along a straight track is compared. A flag is dropped to indic... show full transcript
Worked Solution & Example Answer:A driver is road-testing two minibuses, A and B, for a taxi company - AQA - A-Level Maths Pure - Question 15 - 2018 - Paper 2
Step 1
Find the time taken for A to travel 100 metres.
Answer
To find the time taken for minibus A to cover 100 metres, we first need to calculate the displacement using the given acceleration:
The acceleration of A is given by:
We find velocity by integrating acceleration:
v = rac{d}{dt} ext{(acceleration)} = rac{d}{dt} (0.138 t^2) = 0.276 t
Next, we find the displacement by integrating velocity:
s = rac{d}{dt} ext{(velocity)} = rac{d}{dt} (0.276 t) = 0.138 t^2
Since A travels 100 metres, we set:
Solving for t gives:
t^3 = rac{100}{0.138} \ ext{ and } \ t = rac{100}{0.138} \ = 3.261 \ s
Thus, the time taken for A to travel 100 metres is approximately 3.261 seconds.
Step 2
Find the time taken for B to travel 100 metres.
Answer
To calculate the time taken for minibus B to cover 100 metres, we again integrate the acceleration function:
The acceleration of B is given by:
We first find the velocity:
v = rac{d}{dt} ext{(acceleration)} = rac{d}{dt} (0.024 t^3) = 0.072 t^2
Next, we find the displacement:
s = rac{d}{dt} ext{(velocity)} = rac{d}{dt} (0.072 t^2) = 0.024 t^3
Setting the displacement equal to 100 metres, we find:
Solving for t gives:
t^4 = rac{100}{0.024} \ ext{ and } \ t = rac{100}{0.024} \ = 9.642 \ s
Thus, the time taken for B to travel 100 metres is approximately 9.642 seconds.
Step 3
Step 4
Explain why the company may, in reality, make the wrong decision.
Answer
The models assume that both minibuses start moving immediately when t = 0. However, in practical situations, driver reaction times and external factors can delay the start. If Driver B's reaction time is greater than Driver A's, it could result in minibus B being able to cover 100 metres faster overall, despite its longer modelled time. Therefore, real-world conditions may cause the company to make a decision that doesn't reflect the actual performance of the minibuses.