Modigliani
New member
- Joined
- Dec 20, 2008
- Messages
- 6
Hi. My teacher has assigned a series of problems relating to optimization. One of them is his weird version of the usual optimized pipe laying between land and water problem. I found a couple examples with the search feature, but I'm not sure that I did my problem right. This is the wording:
Points A and B are separated by a portion of a swamp. Point B is a small dry island in the middle of the swamp, and is located 10 mi east of Point C. Point A is located 16 mi south of Point C on the same edge of the swamp. A road is to be built connecting Points A and B. The price of building a road through the swamp is 2 times that of building a road along its edge. How far south of Point C should the road enter the swamp?
First, I asked him about the dimensions of the swamp to the north and east, and he just told me to assume it continues on "beyond the realm of the problem." Basically, the swamp is a replacement for the ocean in the pipe from the oil well to oil refinery problem. I made a simple diagram:
k = cost per land unit
C(x) = k*(16-x) + 2k*sqrt(x^2 + 100)
= 16k - kx + 2k*(x^2 + 100)^(1/2)
C' = -k + 2kx/(x^2 + 100)^(1/2)
Set k equal to 1 and solve for zeros.
1 = 2x/(x^2 + 100)^(1/2)
2x = (x^2 + 100)^(1/2)
4x^2 = x^2 + 100
3x^2 = 100
x^2 = 10/3
x = sqrt(10/3)
Have I finished? I did this using the method I found here and with google, but I don't really understand it. Thanks for any help.
Points A and B are separated by a portion of a swamp. Point B is a small dry island in the middle of the swamp, and is located 10 mi east of Point C. Point A is located 16 mi south of Point C on the same edge of the swamp. A road is to be built connecting Points A and B. The price of building a road through the swamp is 2 times that of building a road along its edge. How far south of Point C should the road enter the swamp?
First, I asked him about the dimensions of the swamp to the north and east, and he just told me to assume it continues on "beyond the realm of the problem." Basically, the swamp is a replacement for the ocean in the pipe from the oil well to oil refinery problem. I made a simple diagram:
k = cost per land unit
C(x) = k*(16-x) + 2k*sqrt(x^2 + 100)
= 16k - kx + 2k*(x^2 + 100)^(1/2)
C' = -k + 2kx/(x^2 + 100)^(1/2)
Set k equal to 1 and solve for zeros.
1 = 2x/(x^2 + 100)^(1/2)
2x = (x^2 + 100)^(1/2)
4x^2 = x^2 + 100
3x^2 = 100
x^2 = 10/3
x = sqrt(10/3)
Have I finished? I did this using the method I found here and with google, but I don't really understand it. Thanks for any help.
