# Optimization

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This is one optimization problem that I just cant figure out. I'll post what I have...

A hiker at point A on a straight road wants to reach, in the shortest time, a point B located 6 miles from the road and 10 miles from point A. The hiker's speed on the paved road is 4 mph and only 2 mph off the road. How far should he continue on the road before heading in a straight line for the point B?

I am pretty sure I would have to use the pythagorean theorem because if you draw the problem you get a triange with two sides given. Plus I denoted $dr/dt$ to be the speed on the road which is 4 mph, and $do/dt$ to be the speed off road which is 2 mph. I just don't know how to find my objective function. Any help would be appreciated.

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You're on the right track. With your drawing, split the problem in 2 parts using an intermediate point P. Express the distance as a function of the distance to P by using the pyth. theorem on the triangle, compute the time necessary to do both parts with the given speeds and then add those two parts.

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right... at want point p would the hiker be able to travel the distance the fastest?

I got 4.3

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I can't check your answer since the question isn't 100% clear to me. Is B located 6 miles under the road, vertically? So the distance between B and the road is 6. And the 10 miles, is that the (straight) distance between A and B or between A and the point on the road which is above B?

A___________

| 6m

B

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