2024 DHS Promo Q12

Alan wants to sign up for a triathlon competition which requires him to swim for \(1.5\) km, cycle for \(30\) km and run for \(10\) km. He plans a training programme as follows: In the first week, Alan is to swim \(400\) m, cycle \(1\) km and run \(400\) m. Each subsequent week, he will increase his swimming distance by \(50\) m, his cycling distance by \(15\%\) and his running distance by \(r\%\).

1. Given that Alan will run \(829.44\) m in Week \(5\), show that \(r = 20\). Hence determine the distance that Alan will run in Week \(20\), giving your answer to the nearest km.[2]
2. Determine the week when Alan first achieves the distances required for all three categories of the competition.[3]

During a particular running practice, Alan plans to run \(q\) metres in the first minute. The distance he covers per minute will increase by \(80\) m for the next four minutes. Subsequently, he will cover \(6\%\) less distance in a minute than that in the previous minute.

3. Show that the distance, in metres, Alan will cover in the sixth minute is \(0.94q + 300.8\) and hence find the minimum value of \(q\), to the nearest metre, such that he can eventually complete \(10\) km.[4]

While training, Alan suffers from inflammation and needs medication. The concentration of the medicine in the bloodstream after administration can be modelled by the recurrence relation

\[C_{n+1} = C_{n} \mathrm{e}^{-\left(p + \frac{1}{n+100}\right)},\]

where \(n\) represents the number of complete hours from which the medicine is first taken and \(p\) is the decay constant.

4. The dosage of the medicine prescribed for Alan is \(200\) mg and the concentration of the medicine dropped to approximately \(168\) mg one hour later. It is given that his pain will be significant once the concentration of the medicine falls below \(60\) mg. Determine after which complete hour he would feel significant pain and should take his medicine again.[3]