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Thepiman
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#1
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#1
A pipeline needs to be laid between an offshore oil rig (A) to the refinery (B) on the coast (length NB). It costs 5 times as much to lay a mile of underwater pipeline as it does to lay a mile of pipeline on land.


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The distance AN is 50 km and distance NB is 100 km. The plan is to lay the pipeline as a straight stretch AM underwater followed by a straight stretch MB on land as above. Determine where point M should be if the pipeline is to be laid as cheaply as possible?
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Smaug123
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#2
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#2
(Original post by Thepiman)
A pipeline needs to be laid between an offshore oil rig (A) to the refinery (B) on the coast (length NB). It costs 5 times as much to lay a mile of underwater pipeline as it does to lay a mile of pipeline on land.


REFER TO ATTACHED IMAGE


The distance AN is 50 km and distance NB is 100 km. The plan is to lay the pipeline as a straight stretch AM underwater followed by a straight stretch MB on land as above. Determine where point M should be if the pipeline is to be laid as cheaply as possible?
Suppose M were laid at point k. How much does the route cost? Then minimise that for k.
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Necrosyrtes
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#3
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#3
(Original post by Thepiman)
A pipeline needs to be laid between an offshore oil rig (A) to the refinery (B) on the coast (length NB). It costs 5 times as much to lay a mile of underwater pipeline as it does to lay a mile of pipeline on land.


REFER TO ATTACHED IMAGE


The distance AN is 50 km and distance NB is 100 km. The plan is to lay the pipeline as a straight stretch AM underwater followed by a straight stretch MB on land as above. Determine where point M should be if the pipeline is to be laid as cheaply as possible?
Find expressions for the length of each section of pipe, and use them to find an equation for price, then use calculus to minimise it.
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WinstonO'Brien
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#4
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#4
Surely M=B LOL
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Necrosyrtes
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#5
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#5
(Original post by WinstonO'Brien)
Surely M=B LOL
Why would that be the case?
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WinstonO'Brien
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#6
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#6
(Original post by Necrosyrtes)
Why would that be the case?
Idk just a guess.

You get a nasty equation when I did it, that: 0=50Sinx(secx)^2-50Cosx
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Necrosyrtes
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#7
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#7
(Original post by WinstonO'Brien)
Idk just a guess.

You get a nasty equation when I did it, that: 0=50Sinx(secx)^2-50Cosx
You don't have to even think about angles - you have a Pythagorean triangle and a straight line.

Potentially confusing 'hint':
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Maybe think of N as an origin?
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WinstonO'Brien
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#8
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#8
(Original post by Necrosyrtes)
You don't have to even think about angles - you have a Pythagorean triangle and a straight line.

Potentially confusing 'hint':
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Maybe think of N as an origin?
meh im stupid lol

how do you do it?
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interstitial
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#9
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#9
(Original post by WinstonO'Brien)
meh im stupid lol

how do you do it?
Left side is 50km.

Bottom side is k

Use Pythagoras in the right angled triangle formed.

Other side is 100-k.

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Necrosyrtes
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#10
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#10
(Original post by WinstonO'Brien)
meh im stupid lol

how do you do it?
Think of it like this:
A = (0, 50)
N = (0, 0)
M = (M, 0)
B = (100, 0)

Now you can find a general expression in terms of M for AM using Pythagoras's theorem (triangle ANM), and another expression in terms of M for the length of BM. Use these to set up an equation for P in terms of M and minimise it.
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WinstonO'Brien
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#11
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#11
(Original post by Necrosyrtes)
Think of it like this:
A = (0, 50)
N = (0, 0)
M = (M, 0)
B = (100, 0)

Now you can find a general expression in terms of M for AM using Pythagoras's theorem (triangle ANM), and another expression in terms of M for the length of BM. Use these to set up an equation for P in terms of M and minimise it.
I appreciate the help but I'm hopeless when it comes to this stuff, thanks a lot though
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Necrosyrtes
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#12
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#12
(Original post by WinstonO'Brien)
I appreciate the help but I'm hopeless when it comes to this stuff, thanks a lot though
Ok then. Let the length NM be equal to m. Now, what is the length of the line AM? Use pythagoras's theorem. You should get an answer in terms of m.

The line NB is 100 km. What is the length of BM in terms of m, given that NM = m and NB = NM+MB?
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WinstonO'Brien
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#13
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#13
(Original post by Necrosyrtes)
Ok then. Let the length NM be equal to m. Now, what is the length of the line AM? Use pythagoras's theorem. You should get an answer in terms of m.

The line NB is 100 km. What is the length of BM in terms of m, given that NM = m and NB = NM+MB?
i have given up, really appreciate your time to reply but i just can't do maths
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