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

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 time taken for the first 6 kilometres, we start with the constant pace for the first 4 kilometres:

Time for first 4 km:

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

Next, we calculate the time for the 5th and 6th kilometre considering the 5% increase:

Time for 5th km:

$6 \text{ minutes} \times 1.05 = 6.3 \text{ minutes}$
Time for 6th km:

$6.3 \text{ minutes} \times 1.05 = 6.615 \text{ minutes}$

Now, summing these times:

Total time = 24 minutes + 6.3 minutes + 6.615 minutes = 36.915 minutes

Converting to minutes and seconds:

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

Thus, the estimation is verified.

Step 2

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

To determine the time for the rth kilometre after the 4th, we see that each subsequent kilometre takes 5% longer than the previous one. Thus, for r ≥ 5:

The formula becomes:

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

This shows that her estimated time for each kilometre follows this geometric sequence.

Step 3

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

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Answer

To find the total time for the 20 kilometre race:

Time for the first 6 km is 36.915 minutes as calculated earlier.
For the remaining 14 km, using the sum of a geometric series:

From the 5th km to the 20th km:

Total time = 36.915 minutes +

$\sum_{k=5}^{20} 6 \times 1.05^{k-4} = 6 \times (1.05^1 + 1.05^2 + ... + 1.05^{16})$

Using the formula for a finite geometric series:

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

Where:

a is the first term (6*1.05),
r is the common ratio (1.05), and
n is the number of terms (16):

Thus, using this formula, the calculation gives approximately:

Total time is approximately 173 minutes and 3 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

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

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

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