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 1

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 1

Step 1

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

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Answer

To find the time taken to run the first 6 kilometres, we calculate the time for each of the first 4 kilometres and then add the time for the 5th and 6th kilometres, which is calculated with the 5% increase.

Calculate time for the first 4 km:
Each kilometre takes 6 minutes, so:

$4 ext{ km} = 4 imes 6 ext{ minutes} = 24 ext{ minutes}$
Calculate time for the 5th kilometre:
Time for the 5th km is:

$6 ext{ minutes} + 5\ ext{%} = 6 imes 1.05 = 6.30 ext{ minutes}$
Calculate time for the 6th kilometre:
Time for the 6th km is:

$6.30 imes 1.05 = 6.615 ext{ minutes}$
Total time for first 6 km:

$24 + 6.30 + 6.615 = 36.915 ext{ minutes}$

Converting 36.915 minutes to minutes and seconds:
$36 ext{ minutes} + 0.915 imes 60 seconds = 36 ext{ minutes } 55 ext{ seconds}.$

Thus, the estimated time to run the first 6 km is 36 minutes 55 seconds.

Step 2

(b) 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

To formulate the expression for the time taken to run the rth kilometre:

Base Time for 5th km: The time for the 5th km is calculated as above:

$6 imes 1.05^{(5-4)} = 6 imes 1.05^1 = 6 imes 1.05$
General Formula:
For the rth kilometre where r is 5 or greater:

$T_r = 6 imes 1.05^{(r-4)}$
Thus, the expression is validated for r between 5 and 20.

Step 3

(c) 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 20 km race:

Time for first 4 km:
- Time for the first 4 km is already calculated: $24 ext{ minutes}$
Calculate time for remaining 16 km:
The total time for the remaining 16 km can be represented as a sum of a geometric sequence:

$ext{Time} = 6 + 6 imes 1.05 + 6 imes 1.05^2 + ext{...} + 6 imes 1.05^{15}$

This is a geometric series with:
- First term (a) = 6
- Common ratio (r) = 1.05
- Number of terms (n) = 16
Using the formula for the sum of a geometric series:

$S_n = a rac{(r^n - 1)}{r - 1}$

Where: $S_{16} = 6 rac{(1.05^{16} - 1)}{1.05 - 1}$

After calculating, the total time:

$S_{16} ext{ (remaining 16 km)} = 30.427 ext{ minutes}$
Total estimated time:

$ext{Total Time} = 24 + S_{16} = 24 + 30.427 = 54.427 ext{ minutes}$ Converting to minutes and seconds gives: $54 ext{ minutes } 25.62 ext{ seconds}.$
Therefore, the estimated total time to complete the race is approximately 173 minutes and 3 seconds.

A competitor is running a 20 kilometre race - Edexcel - A-Level Maths Pure - Question 13 - 2019 - Paper 1

Worked Solution & Example Answer:A competitor is running a 20 kilometre race - Edexcel - A-Level Maths Pure - Question 13 - 2019 - Paper 1

(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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