6th CanMO 1974 Problem 7 Bus A leaves the terminus every 20 minutes, it travels a distance 1 mile to a circular road of length 10 miles and goes clockwise around the road, and then back along the same road to to the terminus (a total distance of 12 miles). The journey takes 20 minutes and the bus travels at constant speed. Having reached the terminus it immediately repeats the journey. Bus B does the same except that it leaves the terminus 10 minutes after Bus A and travels the opposite way round the circular road. The time taken to pick up or set down passengers is negligible. A man wants to catch a bus a distance 0 < x < 12 miles from the terminus (along the route of Bus A). Let f(x) the maximum time his journey can take (waiting time plus journey time to the terminus). Find f(2) and f(4). Find the value of x for which f(x) is a maximum. Sketch f(x).
The bus takes 20 mins for 12 miles or 5/3 min/mile. For x = 3. So, summarising: f(x) = 10 + 5x/3 for 0 ≤ x <= 1 20 + 5x/3 for 1 < x <= 3 30 - 5x/3 for 3 < x <= 6 10 + 5x/3 for 6 < x <= 9 40 - 5x/3 for 9 < x < 11 30 - 5x/3 for x ≥ 11 f(2) = f(4) = 23 1/3. The maximum value of 25 minutes occurs at x = 3 and x = 9. 6th CanMO 1974 © John Scholes jscholes@kalva.demon.co.uk 10 June 2002 Last corrected/updated 1 Mar 03