as you figured, the time t is
t = (300-x)/(160) + √(800^2 + x^2)/70
dt/dx =
160x - 70√(800^2 + x^2)
-----------------------------
70*160√(800^2 + x^2)
dt/dx=0 when
160x = 70√(800^2 + x^2)
25600x^2 = 56000000 + 4900x^2
x = 389.23
Hmm. Not what I had expected. How about if we let x be the distance run along the trail. Then
t = x/160 + √(800^2 + (300-x)^2)/70
Hmmm. That gives -89.23, the overrun in the above derivation. How strange. I expect the actual solution is
x = 300-89.23 = 210.77, since the distance traveled in the woods is the same, but the trail distance is shorter.
Maybe you can see where the spurious sign change creeps in.
This is the question: A man is running through the woods to a check point as fast as possible. He can get there by traveling east along the trail for 300 meters, and then north through the woods for 800 meters. He can run 160 meters per minute along the trail, but only 70 meters per minute through the woods. Running directly through the woods toward the checkpoint minimizes the distance, but he will be going slowly the whole time. He could instead run for a period along the trail before entering the woods, maximizing the total distance but minimizing the time in the woods. Perhaps the fastest route is a combination. Find the path that will get him to the tree in the minimum time.
So I know the equation is
(300-x)/(160) + (square root(800^2 + x^2))/70
- x is the distance of the 300 meter path that is not traveled, so the distance traveled is the 300-x
I did the equation, but i ended up with x equaling greater than 300. Did I not have the right equation can someone help?
2 answers
You must have done it correctly, since I also got x = appr 389 m
Since this does not appear to make any real sense, let's investigate the other 2 cases:
1. he runs all the 300 m on the trail then at right angles through the bush
2. he runs all the way through the bush
case1: x = 0 for your equation
Time = 300/160 + 800/70 = 13.3 min
case2: x = 300 in your equation
time = √(300^2 + 800^2) /70 = 12.2 min
suppose we take a route where he runs 200 m along the trail and then through the woods
time = 200/160 + √(100^2 + 800^2)/70 = 12.76 , which is greater than case2
Conclusion: He will not gain any time by doing a partial trail and then angling through the woods.
This is a standard type of question, and in most cases there is a intended solution, but in this case somebody made up the question without really watching the numbers.
Since this does not appear to make any real sense, let's investigate the other 2 cases:
1. he runs all the 300 m on the trail then at right angles through the bush
2. he runs all the way through the bush
case1: x = 0 for your equation
Time = 300/160 + 800/70 = 13.3 min
case2: x = 300 in your equation
time = √(300^2 + 800^2) /70 = 12.2 min
suppose we take a route where he runs 200 m along the trail and then through the woods
time = 200/160 + √(100^2 + 800^2)/70 = 12.76 , which is greater than case2
Conclusion: He will not gain any time by doing a partial trail and then angling through the woods.
This is a standard type of question, and in most cases there is a intended solution, but in this case somebody made up the question without really watching the numbers.
7 answers