Photo AI
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
Answer
To calculate the time taken to run the first 6 kilometres, we consider the first 4 kilometres and then add the time for the 5th and 6th kilometres:
-
For the first 4 kilometres, the time is:
-
The time for the 5th kilometre is:
-
The time for the 6th kilometre is:
-
Now, we add these times: This converts to seconds as follows:
Step 2
Show that her estimated time, in minutes, to run the rth kilometre, for 5 ≤ r < 20, is 6 × 1.05^{r−4}
Answer
For any kilometre run after the first four, the time taken can be expressed as a geometric progression. The time taken for the rth kilometre (where r ≥ 5) is calculated as follows:
Given that the time taken for the 5th kilometre is 6.3 minutes, we can express the time for the rth km as:
This indicates that the time taken increases by 5% for each subsequent kilometre after the fourth.
Step 3
Estimate the total time, in minutes and seconds, that she will take to complete the race
Answer
To estimate the total time for the entire race of 20 kilometres, we break down the time as:
-
Time for the first 6 kilometres:
-
For the remaining 14 kilometres (from 7 to 20): We calculate:
= 6 × rac{1.05^{16} - 1}{1.05 - 1}$$ Evaluating this gives us the time spent on the remaining kilometres: $$ 6 × rac{1.05^{16} - 1}{0.05} ext{ minutes}$$ -
Total time is:
-
The total time computed comes out to approximately 173 minutes and 3 seconds.