RI APGP Tutorial Q9

The coach of Besto running club designed a training programme such that runners begin with a \(400\) m run on the first training session. On each subsequent session, the distance covered is \(250\) m more than the distance covered on the previous session.

1. Find the minimum number of sessions required for runners from Besto on the training programme to run at least \(20\) km in a training session.[3]

For another group of runners in Besto, a circuit training exercise was designed to build up their stamina.

In this exercise, this group of runners from Besto run from a starting point \(O\) to and from a series of points, \({{P}_{1}}\), \({{P}_{2}}\), \({{P}_{3}}\),… , increasingly far away in a straight line. In the exercise, they start at \(O\) and run stage 1 from \(O\) to \({{P}_{1}}\), and back to \(O\), then stage 2 from \(O\) to \({{P}_{2}}\), and back to \(O\), and so on.

The distances between the points are such that \({OP_1 = 50}\)m, \({P_1P_2 = 100}\)m, \({P_2P_3 = 300}\)m and \({P_nP_{n+1} = 3P_{n-1}P_n}\) (see Fig. 1).

2. Find an expression for the distance run by a runner from Besto who completes \(n\) stages of the circuit training exercise.[3]
3. Hence, find the distance from \(O\) and the direction of travel, of a runner from Besto undergoing the circuit training exercise after he has run exactly \(42\) km.[3]

Another running club, Choco, designed a different training programme. The runners in Choco began with running \(400\) m on the \({{1}^{\mathrm{st}}}\) session. On each subsequent session, the distance covered was increased by \(10%\) of the distance covered on the previous session. From the \({{11}^{\mathrm{th}}}\) session onwards and for all subsequent sessions, the distance covered was increased by \(r%\) of the distance covered on the previous session.

4. Given that the runners from Choco club covered at least \(20\) km on the \({{70}^{\mathrm{th}}}\) training session, find the range of values of \(r\).