They begin by letting a typical rider's on-road speed be a and her off-road speed be b. The rider's total time is then given by the function
| Fran and Rebecca have several friends also riding in the same race. Rather than doing the same calculation over and over again for each of them, they decide to find a formula that works for all riders. |
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They begin by letting a typical rider's on-road speed be a and her off-road speed be b. The rider's total time is then given by the function
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| 9. Finish their calculation by modifying the calculation on the previous page. The optimal turnoff distance for this typical rider is in terms of a and b. (Remember when you differentiate that a and b are just constants like 15 and 25.) Check your formula by seeing if it works for Fran and Rebecca's turnoff distances or by checking sample cases with the analyser. |
| 10. What formula did they get for the sine of this typical rider's optimal turnoff angle for the reversed ride? Check that the formula works by checking that it agrees with your calculations for Fran and Rebecca's turnoff angles, or by checking sample cases with the analyser. |
| (Bonus problem) You should have found a very simple formula for the sine of turnoff angle for the reverse ride: it looks like the numbers 7 and 16 (which describe the location of the original end point relative to the start point) don't have any effect on the answer. Prove that this is not just a coincidence by re-doing the problem with the numbers 7 and 16 replaced by arbitrary constants (like you did with a and b) and showing that the formula for the sine of the turnoff angle turns out not to contain those constants. In other words, the turnoff angle is the same no matter where the end point is located relative to the start. |