A competitor is running a 20 kilometre race. She runs each of the first 4 kilometres at a steady pace of 6 minutes per kilometre. After the first 4 kilometres, she begins to slow down. In order to estimate her finishing time, the time that she will take to complete each subsequent kilometre is modelled to be 5% greater than the time that she took to complete the previous kilometre. Using the model, (a) show that her time to run the first 6 kilometres is estimated to be 36 minutes 55 seconds. (b) show that her estimated time, in minutes, to run the r th kilometre, for 5 ≤ r < 20, is $$6 \times 1.05^{r-4}$$ (c) estimate the total time, in minutes and seconds, that she will take to complete the race.

A-Level Edexcel Maths Pure Basic Trigonometry: A competitor is running a 20 kil

A competitor is running a 20 kilometre race - Edexcel - A-Level Maths Pure - Question 12 - 2019 - Paper 2

Question 12

A competitor is running a 20 kilometre race. She runs each of the first 4 kilometres at a steady pace of 6 minutes per kilometre. After the first 4 kilometres, she ... show full transcript

Worked Solution & Example Answer:A competitor is running a 20 kilometre race - Edexcel - A-Level Maths Pure - Question 12 - 2019 - Paper 2

Step 1

show that her time to run the first 6 kilometres is estimated to be 36 minutes 55 seconds

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Answer

To calculate the total time for the first 6 kilometres, we break it down as follows:

The time taken for the first 4 kilometres is: $4 \text{ km} \times 6 \text{ min/km} = 24 \text{ minutes}$
The time taken for the 5th kilometre: $6 \text{ min} \times 1.05 = 6.3 \text{ minutes}$
The time taken for the 6th kilometre: $6.3 \text{ min} \times 1.05 = 6.615 \text{ minutes}$
Therefore, the total time for the first 6 kilometres is: $24 + 6.3 + 6.615 = 36.915 \text{ minutes}$

Which converts to 36 minutes 55 seconds.

Step 2

show that her estimated time, in minutes, to run the r th kilometre, for 5 ≤ r < 20, is 6 × 1.05^{r-4}

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Answer

For the 5th kilometre, the time taken can be represented as: $6 \text{ min} \times 1.05^{1} \text{ (for the 5th km)}$

For the 6th kilometre: $6 \text{ min} \times 1.05^{2} \text{ (for the 6th km)}$

Thus, the general formula for the r th kilometre (for 5 ≤ r < 20) can be expressed as:

$6 \times 1.05^{r-4}$

Step 3

estimate the total time, in minutes and seconds, that she will take to complete the race

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Answer

To estimate the total time for the entire race:

The total time for the first 6 kilometres is $36.915 \text{ minutes}$ .
For the remaining kilometres (from 7 to 20), we set up a series: $T = 24 + \sum_{k=4}^{16} 6 \times 1.05^{k}$
Recognizing the series is a geometric series: $T = 24 + 6 \times \frac{1.05(1.05^{16}-1)}{1.05-1}$
Calculating this gives us approximately 173 minutes 3 seconds, rounding appropriately.

A competitor is running a 20 kilometre race - Edexcel - A-Level Maths Pure - Question 12 - 2019 - Paper 2

Worked Solution & Example Answer:A competitor is running a 20 kilometre race - Edexcel - A-Level Maths Pure - Question 12 - 2019 - Paper 2

show that her time to run the first 6 kilometres is estimated to be 36 minutes 55 seconds

show that her estimated time, in minutes, to run the r th kilometre, for 5 ≤ r < 20, is 6 × 1.05^{r-4}

estimate the total time, in minutes and seconds, that she will take to complete the race

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