An Oil Refinery and where to place the pipeline?

SydneyRoverP6B

SydneyRoverP6B

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An oil refinery is located on the north bank of a straight river that is 2km wide. A pipeline is to be constructed from the refinery to the storage tanks located on the south bank of the river 6km east of the refinery.
The cost of laying pipe is $400,000/km over land to a point 'P' on the north bank and $800,000/km under the river to the tanks. To minimise the cost of the pipeline, where should 'P' be located?

Ron.
 
Well... This is as far as i've got. I've cheated on the differentiation (must get back into that)

The point P is at 1.154 km along the lower bank i think...

I did this by saying the amount it costs to build is:
2400000-400000*2Tan x +800000 *2/cos x.
or for value of x we can factor down to 3-tan x + 2/cos x.
i.e. Cost of total distance along bank, - excluded section + river crossing section

I graphed this - value for zero between 0 and PI/2 is 0.5235

Tan x = opp / adj so Opp = Tan x * adj giving Opposite of 1.1544 km.

Thoughts? Seems a little lower than i expected!

Rich.
 
I get that P should be situated at 4.8453km along the north bank, the pipe then running 2.3094km from point P under the river to the tanks, the total cost being $3.786 million.

Please feel free to say I'm wrong if I am... and please feel free to admire me if I'm right! :p
 
Ah.... I've put the pipeline on the south bank rather than the north - hence my answer is 6km minus your answer 8)

Rich.
 
Hello Rich arnd Darth,

:D Yes indeed,...Point 'P' is located a distance 6 - 2/3^1/2 km ( approx 4.85km to two dp ) east of the refinery on the opposite side to the storage tanks.

Top marks gents,... :D

Ron.
 
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