Optimization Prob: least cost for cable across river

[truthfulone]

A power house, P, is on one bank of a straight river 200 m wide, and a factory, F, is on the other bank 400 m 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 so the cost is minimized?

Here is an image if I understand the question correctly:

Please help me out. Thanks.

 

[galactus]

The thing to do is draw a point on the other end of the river and let's call it Q. That is the point where the cable will come out of the water and go along the bank of the river to the factory.

By Pythagoras, the cost of the cable underwater, from P to Q, given by \(\displaystyle 12\sqrt{200^{2}+x^{2}}\)

The cost over land, from Q to F, is given by \(\displaystyle 6(400-x)\)

So, the total cost is given by \(\displaystyle 12\sqrt{40,000+x^{2}}+6(400-x)\)

Now, differentiate, set to 0 and solve for x. Then you can plug that in to find the distances you need.

 

[truthfulone]

thanks for your response.

I have differentiated the total cost equation. For x I get 115.47 m.

Am I on the right path? Please help me. Thanks.

 

[galactus]

Yes, good, that's right. Now use that to find the distances and thus the cost.

 

[truthfulone]

Wow. Thanks.
I found the underwater distance to be 230.94. And the ground distance to be 284.53 m.

Then the total cost comes $4478.46.

I hope I am right. Thanks.