A hunter is at a point along a river bank. He wants to get to his cabin, located 3 miles north and 8 miles west. He can travel 5mph along the river but only 2mph on this very rocky land. How far upriver should he go in order to reach the cabin in minimum time ?
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The vital piece of information that isn't given in this problem is the orientation of the river. I'm going to assume the river runs north-south... so that when the hunter goes "upriver", he's heading north along the river bank.
This question was tricky... you have to kind of think in reverse of what you'd normally do to set up the problem. We're not looking for a function of time here... we're looking for a function that yields the time. The key is that d = rt implies that d/r = t.
So let Point S be the hunter's starting point. The hunter is going to walk upriver (north) for awhile to Point A, then veer off and make a bee line toward the cabin, which we'll call Point C. Finally, let Point B be the point on the river due east of the cabin. Note that triangle ABC is a right triangle.
Let ||AB|| = d; then ||SA|| = 3 - d. You already know ||BC|| = 8, so all that's left is to determine ||AC||, which is the hypoteneuse of the triangle.
||AC||^2 = ||AB||^2 + ||BC||^2
||AC||^2 = d^2 + 8^2
||AC||^2 = d^2 + 64
||AC|| = (d^2 + 64)^(1/2)
Now you can create a function T(d) that gives the total time of the hunter's journey.
T(d) = time it takes to travel ||SA|| + time it takes to travel ||AC||
T(d) = ||SA||/rate of travel upriver + ||AC||/rate of travel over rocks
T(d) = (3-d)/5 + (d^2 + 64)^(1/2)/2
To determine the minimal time, use the First Derivative Test to find the value of d for which T(d) is minimized.
T'(d) = -1/5 + d/[2(d^2 + 64)^(1/2)]
0 = -1/5 + d/[2(d^2 + 64)^(1/2)]
1/5 = d/[2(d^2 + 64)^(1/2)]
1/25 = d^2 / 4(d^2 + 64) <----- Square each side of equation.
1/25 = d^2 / (4d^2 + 256)
4d^2 + 256 = 25d^2 <----------- Cross multiply.
256 = 21d^2
256/21 = d^2
16/sqrt(21) = d
||SA|| = 3 - d = 3 - 3.49 = -.49 miles. Oops...
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The above answer is correct--at least in terms of the function, though the solution is out-of-range for the problem--but remember I assumed the river was running north-south. Let's see what happens if I assume the river runs east-west instead, with "upriver" being toward the west.
S is the starting point. The hunter walks west to point A then veers off toward the cabin at Point C. Point B is the point on the river directly south of the cabin.
||AB|| = d
||SA|| = 8 - d
||BC|| = 3
||AC|| = (d^2 + 9)^(1/2)
T(d) = (8 - d)/5 + (d^2 + 9)^(1/2)/2
T'(d) = -1/5 + d/[2(d^2 + 9)^(1/2)]
0 = -1/5 + d/[2(d^2 + 9)^(1/2)]
1/5 = d / [2(d^2 + 9)^(1/2)]
1/25 = d^2 / [4(d^2 + 9)]
4(d^2 + 9) = 25d^2
4d^2 + 36 = 25d^2
36 = 21d^2
36/21 = d^2
6/sqrt(21) = d
||SA|| = 8 - d = 8 - 1.3 = 6.7... so the hunter should walk west for about 6.7 miles before heading to the cabin.