A company wishes to run a utility cable from point A on the shore to an installation at point B on the island. The island is 6 miles from the shore. It costs $400 per mile to run the cable on land and $500 per mile underwater. Assume that the cable starts at A and runs along ths shoreline, then angles and runs underwater to the island. Find the point at which the line should begin to angle in order to yield the minimum total cost.
We also did one like this in class and I have the notes that I can show you, maybe it will help:
There's a guy who wants to get from point A to point B. Straight along a path is 300m and he can travel at 160m/min. The rest is woods at which he can travel at 70m/min and perpendicular from the path to point B is 800m. What route would minimize the amount of time?
D = R*T
T = D/R
E = 300m at 160 mpm = 18 min
N = 800m at 70 mpm = 11.42 min
So the total time would be straight along the path for 18 minutes, then up through the woods for 11.42 minutes...
So what would minimize the amount of time?
Pythagorean theorum: A^2 + B^2 = C^2
So: what distance on the path (x)^2 plus distance up through the woods (800^2) = d^2
So D = SQRT (x^2 + 800^2)
....Total time on path is 300 (length of path) - x @ 160 mpm
Total Tme is: 300-x/160 + SQRT(x^2+800^2)/70
T'(x) = ... etc
set = 0 for critical points
get crazy number
get 389 meters however this is not in the domain of [0, 300]
so minimum has to occur at an endpoint.
t(0) = 13.9 min
t(300) = 12.83...
so his best bet is traveling entirely through the woods
Let the required point along the shoreline be P.
Let dPC = x so that dAP = 9-x.
From Pythagoras: .
Cost ($) .
Differentiate the cost function with respect to x (use the chain rule), put the derivative equal to zero and solve for x:
It's left to you to finish solving for x and then to test the nature of the solutions. You want the solution that corresponds to a minimum turning point. Call it x = p. Then the value of x that minimises the cost will be one of the following: p, 0, 9.
It's left for you to think about where 0 and 9 come from and why they should be considered.
So I solved for x, but I'm still confused where to take it from there. I know that x = 2.6666 or 2 2/3...
i figured out that the distance between A and B when taking a direct route diagonially ( the C^2 in the pythagorean theorum) is 10.81
Using endpoint 0 ( 9 on land, 6 underwater ) you get 3600 + 3000 = $6600
Using endpoint 9 ( all underwater) or 10.81 under ($500) you get $5408.33
Therefore I know it is cheaper to go completely underwater than from A to C then from C to B...
however I'm not sure about the middle ones... :-\