pipeline word problem

[dear2009]

Hey everybody,



I am trying to solve this word problem through the derivative but I cant seem to get the right answer even though I did the right method

Here is the problem
The task is to construct the minimum-cost pipeline from an offshore platform to an onshore refinery. The refinery is located 10 miles up the coast from the platform, and the platform is located 2 miles off the coast, as shown in the diagram below. Assuming it costs $18K per mile to build the pipeline on the land, and $30K per mile to construct the pipeline offshore, what is the minimum cost for the project?

Here is what I did

(square root of 4 + x^2)(30 + (2 - x) 18))
(square root of 4 + x^2)(30 + 36 - 18x)
C'(x) = 0
C ' (x) = 1/2 (4 + x^2)^ -1/2 (2x) (30) - 18
(30x) /( square root of 4 + x^2) - 18 = 0
(30x) /( square root of 4 + x^2) = 18

I eventually got 25x^2/9 = 4 + x^2
x = 3/2

25x^2 - 9x^2 / 9 = 4

when I plugged this in to the (square root of 4 + x^2)(30 + (2 - x) 18))
I got 84 (from 75 + 9, on the last step of plugging in)
but the answer is supposed be around 228-233

Can anybody help me?

[galactus]

The diagram you have appears to have different costs than what is in the problem statement.

The cost on land is

The cost underwater is

Total cost is

Differentiate and get

Set to 0 and solve for x:





Square both sides and simplify:





Plug this back into the total cost formula and get C=228