| Fran is riding in a mountain bike race. The end point of the race is 16 km east of the start and 7 km north of a road running east from the start. Fran will ride the first part of the race on the road and then turn off and head directly towards the end. She can ride her bike at 25 km/hr on the road and 15 km/hr off road. She needs to decide where she should turn off the road to complete the race in the least possible time. |
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The simulation below gives the distance and time for each segment of the ride and the total distance and time (the time for each leg of the trip is found by dividing its distance by the speed). Drag the turnoff point along the road to estimate the position that gives the minimum total time for the ride. Record your answer to this question and to the questions below to compare with your calculated answers from the next page. |
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| 1. Fran's friend Rebecca is not as fit as Fran, and can ride only 20 km/hr on the road and 10 km/hr off road. Adjust the speeds in the analyser to estimate where Rebecca should turn off to minimize her time. |
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2. A false rumour circulated that the end was only 14 km east of the start, not 16 km. Had the rumour been true, what effect would it have had on Fran and Rebecca's turnoff distances? (Adjust the position of the end point in the analyser.) |
| 3. For practice, Fran decides to ride the race backwards, from end to start. This doesn't change her turnoff point (why not?), but presents a problem: she can't see the turnoff point when she starts her ride. She can ride straight in any direction, but needs to know what that direction should be. There's a side road that runs due south from her new start (the old end) to the main road. Use trigonometry and the appropriate distances from the analyser to estimate Fran's turnoff angle (the angle her optimal direction makes with the side road). |
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4. A rider's off-road speed can be hard to determine, since it depends very much on the terrain to be crossed. Suppose Fran's off-road speed is less than she thinks: how will that affect her optimal turnoff point? (Try a "commonsense answer" first; then confirm your answer with the analyser.) What if her off-road speed is more than she thinks? |
| 5. Use the analyser to explore how changing the end point of the race affects the turnoff distance. How does it affect the turnoff angle for the reversed ride? |