Help needed with Related Rates problem!!!

[Guest dash_dj]

This is the question which I cannot do and it's driving me nuts :

 

A power house, P, is on one bank of a straight river 200m wide and a factory , F, is on the other bank 400m downstream from P. The cable has to be taken across the river under water at a cost of $12.00/m. On land the cost is $6.00/m. What path should be chosen to minimize the cost?

 

This may be easy for some but I AM DYING so all help is appreciated!

Thank you in advance

 

-Diz

[Guest dash_dj]

This is the question which I cannot do and it's driving me nuts :

 

A power house, P, is on one bank of a straight river 200m wide and a factory , F, is on the other bank 400m downstream from P. The cable has to be taken across the river under water at a cost of $12.00/m. On land the cost is $6.00/m. What path should be chosen to minimize the cost?

 

This may be easy for some but I AM DYING so all help is appreciated!

Thank you in advance

 

-Diz

[MandrakeRoot]

Ok without loss of generality we can place P in the origin of some grid. Your factory F is then placed at the point (200,400);

Now let C^1([0,1];R) be the space of continuously differentiable functions from [0,1] into R. Then your problem can be formulated as :

[math]\min_{x,y \in C^1([0,1];\mathbb{R})} \int_0^1 \sqrt{12x'(t)^2 + 6y'(t)^2}dt[/math]

 

Mandrake

[MandrakeRoot]

Ok without loss of generality we can place P in the origin of some grid. Your factory F is then placed at the point (200,400);

Now let C^1([0,1];R) be the space of continuously differentiable functions from [0,1] into R. Then your problem can be formulated as :

[math]\min_{x,y \in C^1([0,1];\mathbb{R})} \int_0^1 \sqrt{12x'(t)^2 + 6y'(t)^2}dt[/math]

 

Mandrake

[jordan]

Would anyone mind explaining what's under the "min" part? I understand the rest of the notation, but I don't follow what you're saying with that first part.

[jordan]

Would anyone mind explaining what's under the "min" part? I understand the rest of the notation, but I don't follow what you're saying with that first part.

[bloodhound]

if i read that correctly then it says

 

the minimum of that integral where x and y are continuously differentiable functions from [0,1] to R.

 

Unfortunately i havent done calculus of variations yet, and i cant solve that problem

 

sucks'';;;;;;,,,,.....

[bloodhound]

if i read that correctly then it says

 

the minimum of that integral where x and y are continuously differentiable functions from [0,1] to R.

 

Unfortunately i havent done calculus of variations yet, and i cant solve that problem

 

sucks'';;;;;;,,,,.....

[MandrakeRoot]

Maybe it is better to try to solve the problem in the following way (or at least approximate the solution). Fix any point in the grid (x1,x2). Now calculate the cost to make a straight-line cable from P to this point and then from this point to F.

Depending on your solution you could do the same thing in considering F to be this new point (x1,x2) and/or (x1,x2) the new origin. Ideally you would solve the problem with variable coordinates for F. This could be maybe some easy approximation of the optimal solution.

 

Mandrake

[MandrakeRoot]

Maybe it is better to try to solve the problem in the following way (or at least approximate the solution). Fix any point in the grid (x1,x2). Now calculate the cost to make a straight-line cable from P to this point and then from this point to F.

Depending on your solution you could do the same thing in considering F to be this new point (x1,x2) and/or (x1,x2) the new origin. Ideally you would solve the problem with variable coordinates for F. This could be maybe some easy approximation of the optimal solution.

 

Mandrake