Plug the value of x in the cost function you found at a to find the minimal costs
This is a question out of a textbook and it would be greatly appreciated if you could point me in the right direction. Don't worry, I've done most of it - I just need a little guidance on what to do next.
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An oil platform is located at point O, at sea, 10km from nearest point P on a stretch of straight coastline. The oil refinery is at point R, 16km along the coast from point P (see diagram). A company needs to lay a pipeline from the platform to the refinery (ie. from point O to R). This is to consist of straight line sections. Laying pipeline underwater is 4/3 times as expensive as laying it on land:
(a) If the pipeline reaches the coast at point X (as shown in my diagram) a distance x km from P in direction of R, find an expression for the cost C (in dollars) of laying the pipeline in terms of x, given that laying the pipeline on land has a fixed cost of A dollars per kilometre. Here's what I've done so far (could you check if it's right?):
Using Pythagoras, OX =
(b) Find the cost of the least expensive route for laying the required pipeline. Here's what I've done:
I differentiated it (hopefully I did it correctly):
And simply solved for x (I used my calculator to do this):
Next, I simply plugged this into my diagram to get the required dimensions:
So the required distance from X to R is 4.661 km.
So the required distance from O to X is 15.119 km.
Now what? Is this the answer? Obviously, it looks rather wrong. Am I supposed to put the dollar, A, somewhere? Please help...
Thanks.