calculus word problem

Question

An isolated farm house is located on the bank of a straight canal that is 200 m wide. Electric power for the house must come from a connection that is on the opposite side of the canal, 500 m downstream. If the cost to lay cable underwater is three times the cost of laying on land, what path should be chosen to minimize cost?

Answer 1

Draw a picture!!!

It might look like this:

Then you can write a cost function

Let the cost to run the land portion is 1 unit/meter meaning the cost to run the water portion is 3 units/meter

So the cost equation for the land portion is "1x" or "x"

The cost to run the water portion is "3w" but, using the Pythagorean Theorem, w = √[(200² + (500-x)²]

w = √[40000 + (250000 - 1000x + x²)]

w = √(x² - 1000x + 290000)

So the cost is equation for the water portion is 3√(x² - 1000x + 290000)

The total cost equation then is:

C(x) = x + 3√(x² - 1000x + 290000) or

C(x) = x + 3(x² - 1000x + 290000)^1/2

To minimize, take the derivative and set it equal to zero:

C'(x) = 1 + (1/2)(3(x² - 1000x + 290000)^-1/2(2x - 1000)

0 = 1 + [(6x - 3000)]/[2((x² - 1000x + 290000)^1/2]

This isn't simple to solve but you can subtract 1 from both sides, cross multiply, and square both sides then solve a quadratic. You'll get two answers, one of which must be thrown out as an extraneous solution.