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 rth kilometre, for 5 ≤ r < 20, is 6 × 1.05^{r - 4} (c) estimate the total time, in minutes and seconds, that she will take to complete the race.

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A competitor is running a 20 kilometre race - Edexcel - A-Level Maths Pure - Question 13 - 2019 - Paper 2

Question 13

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 13 - 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 determine the time it takes to run the first 6 kilometres:

For the first 4 kilometres, the time taken is:
- 4 km × 6 min/km = 24 minutes.
For the 5th kilometre:
- Previous time (6 min) × 1.05 = 6.3 minutes.
For the 6th kilometre:
- Previous time (6.3 min) × 1.05 = 6.615 minutes.
Total time for the first 6 kilometres is:

$ext{Total Time} = 24 + 6.3 + 6.615 = 36.915 ext{ minutes}$
Converting 36.915 minutes into minutes and seconds:
- 36.915 minutes = 36 minutes and 55 seconds.

Step 2

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

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Answer

The time taken to run the rth kilometre can be expressed based on the previous kilometre's time:

For r = 5:
- Time = 6 × 1.05^{5 - 4} = 6 × 1.05.
For r = 6:
- Time = 6 × 1.05^{6 - 4} = 6 × 1.05^2.
For r = 7, it continues:
- Time = 6 × 1.05^{7 - 4}.

Thus, the formula for the rth kilometre, where 5 ≤ r < 20, is:

$ext{Time} = 6 × 1.05^{r - 4}$ .

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 whole 20 km race:

The time for the first 6 km is already calculated as 36.915 minutes.
For the remaining 14 km (from 7 to 20), we calculate:

$ext{Total Time} = 24 + ext{Sum from } r = 5 ext{ to } 20 (6 × 1.05^{r - 4}).$
The series can be calculated using the formula for a geometric series:
- Sum = a × (1 - r^n) / (1 - r), where a is the first term, r is the common ratio, and n is the number of terms.
Applying this:

$ext{Total Time} = 24 + 6.3 × rac{1.05^{15} - 1}{1.05 - 1}$
After calculations, we find:
- Total Time = approximately 173 minutes and 3 seconds.

A competitor is running a 20 kilometre race - Edexcel - A-Level Maths Pure - Question 13 - 2019 - Paper 2

Worked Solution & Example Answer:A competitor is running a 20 kilometre race - Edexcel - A-Level Maths Pure - Question 13 - 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 rth 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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