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 $$6 \times 1.05^{r-4}$$ (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 12 - 2019 - Paper 1

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 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 her total time for the first 6 kilometres, we calculate the time for the first 4 kilometres and then add the time for the 5th and 6th kilometres, which are longer due to the 5% increase.

The time for the first 4 kilometres is:

$4 \text{ km} \times 6 \text{ min/km} = 24 \text{ minutes}$

Next, calculate the time for the 5th kilometre:

$\text{5th km time} = 6 \times 1.05^1 = 6.3 \text{ minutes}$

Now, for the 6th kilometre:

$\text{6th km time} = 6 \times 1.05^2 = 6.615 \text{ minutes}$

Adding these times:

$\text{Total time} = 24 + 6.3 + 6.615 = 36.915 \text{ minutes}$

We convert 36.915 minutes to minutes and seconds:

$36.915 \text{ minutes} = 36 \text{ minutes} + (0.915 \times 60) \text{ seconds} = 36 \text{ minutes} 55 \text{ seconds}$

Step 2

(b) show that her estimated time, in minutes, to run the rth kilometre, for 5 ≤ r < 20, is $$6 \times 1.05^{r-4}$$

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Answer

From the previous analysis, the time taken to complete each km increases by 5% from the previous km starting at the 5th km. Therefore, for the 5th km:

$\text{Time for 5th km} = 6 \times 1.05^{1}$

For the 6th km:

$\text{Time for 6th km} = 6 \times 1.05^{2}$

For general $r$ where $r \geq 5$ it becomes:

$\text{Time for rth km} = 6 \times 1.05^{r-4}$

This shows the formula holds for all km from 5 to 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 entire race, we start with the time for the first 6 km and then use the formula for the 14 remaining km:

The time for the first 6 km has already been calculated as 36.915 minutes.

For the remaining km from 7 to 20, we use:

$\text{Total time for } r=7 \text{ to } 20 = \sum_{r=7}^{20} 6 \times 1.05^{r-4}$

This is a geometric series.

Calculating using the series sum formula, we find:

$\text{Time for } 7th \text{ to } 20th km = 6 \times \frac{1.05^{17}-1}{1.05-1}$

Calculating the geometric series gives an estimate of approximately 136.3 minutes.

Adding this to the 36.915 minutes results in a total time of:

$\text{Total} = 36.915 + 136.3 = 173.215 \text{ minutes}$

This converts to approximately 173 minutes and 13 seconds.

A competitor is running a 20 kilometre race - Edexcel - A-Level Maths Pure - Question 12 - 2019 - Paper 1

Worked Solution & Example Answer:A competitor is running a 20 kilometre race - Edexcel - A-Level Maths Pure - Question 12 - 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 $$6 \times 1.05^{r-4}$$

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

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