minimum value calculations

[asroc]

An oil production platform is 9 sqrt3km directly offshore from the nearest point A on a straight coast. It is to be connected by a pipeline to an onshore refinery R at a distance 100km along the straight coastline from A. It costs �1 million per km to lay the pipeline onshore, and �2million per km oto lay it underwater.

By introducing one sensible unknown write an expression for the cost of the pipeline. Find the minimum value of this cost.

(if x is the unknown and y is the cost, find dy/dx=0. Explain why this is a minimum value.)

[crofter]

Let x = distance along the coast where pipe joins "land"
Taking costs in millions = y
Then y = (100 - x) + 2 sqrt(243 + x^2)

So dy/dx = -1 +2x sqrt (243 + x^2) ^ (-1)
Solving dy/dx = 0 for turning values gives x = +/-9
Discard the negative solution and take x = 9 km

So y(min) = 91 + 2sqrt (243 + 81) = 91 + 36 = 127 millions

Check that d2y/dx2 is positive at x = 9 for a local minimum

[Aquagility]

Congratulations, crofter. I could have solved that 60 years ago - but not now, certainly not now.

[crofter]

Aquagility You must have hundreds of stories to tell and one day perhaps I'll get to hear them!
I acquired such skills 45 years ago and have been using them every week since! My abacus is getting a bit rusty nowadays, but can still work out the scores for the Links Games. Cheers Aq.

[crofter]

asroc For the sake of completeness, for d2y/dx2 you need the product rule and the chain rule for differentiation and the result should simplify to
d2y/dx2 = 486/(243 + x^2)^(3/2) > 0 for all real values of x.

Hence x = 9 leads to a minimum

[asroc]

Many thanks to you all. Its amazing once I read how to do it I remembered it all!! Typical!!
Anyway...Thanks again,

asroc