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You’re off-road, in the desert, driving an all-terrain vehicle. You’re $10$ miles off the straight highway, and your destination is $100$ miles ahead, also $10$ miles off the straight highway. You can drive 40 miles per hour off-road, and $80$ miles per hour on the highway. What is the fewest number of minutes in which you can get to your destination?

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  • $\begingroup$ Does 100 miles ahead mean 100 miles in the direction of the highway? $\endgroup$ – f'' Jun 22 '16 at 5:03
  • $\begingroup$ Yeah, it means 100 miles in direction of the highway. $\endgroup$ – Anurag Jain Jun 22 '16 at 5:05
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There are two possibilities: either the shortest way does not require traveling along the highway, or it does. Obviously the shortest route that does not use the highway is a straight line, taking 150 minutes.

To find the shortest route with the highway, we find the critical angle. The ratio of the speeds is 1 to 2, so the critical angle is $\arcsin(\frac{1}{2})=30°$ and we need to meet the road at an angle of $90°-30°=60°$. The same applies for when we leave the road at the end:

The total distance traveled off the road is $2\cdot10\csc(60°)=\frac{40\sqrt3}3$ miles and the total distance traveled along the road is $100-2\cdot10\cot(60°)=100-\frac{20\sqrt3}3$ miles. The total time taken is $15\sqrt3+75\approx101$ minutes.

This is less time than the straight-line route, so it is the answer.

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You can do it in 105 minutes

10 miles to get to the highway = 10 / 40 * 60 = 15 minutes
100 miles on the highway =  100 / 80 * 60 = 75 minutes
10 miles from the highway to the destination = 10 / 40 * 60 = 15 minutes
Total is 105 minutes

Going fully off road would take you 150 minutes (100/40*60) by comparison

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  • $\begingroup$ I guess you need to prove it's minimal. $\endgroup$ – Ankoganit Jun 22 '16 at 5:13

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