what is the cheapest way to lay the pipe?

**bradscoolio** · October 29th 2012, 01:01 AM

Hi guys!

I attached a digram explaining the equation and I'd really like to know how to get the cheapest rout

This question has been pondering in my mind for a long lime and I've never known how to work it out
what is the cheapest way of laying the pipe?
also what is the distance between B & X to get the cheapest price?

Thanks in advance!

Edit: Please keep the distance B to X rounded to 3 decimals as that is the closest meter

**bradscoolio** · October 29th 2012, 01:05 AM

Just by working it out by trial and error got me BX=5.8km and price was $3,165,400.

**MarkFL** · October 29th 2012, 01:26 AM

The distance in km of the underwater portion is:

$\sqrt{x^2+10^2}$

The distance in km of the underground portion is:

$16-x$

And so, the total cost in thousands of dollars is:

$C(x)=190\sqrt{x^2+10^2}+95(16-x)$ where $0\le x\le16$

To find the minimum, we need to differentiate this cost function with respect to $x$ and equate to zero:

$C'(x)=\frac{190x}{\sqrt{x^2+10^2}}-95=\frac{95(2x-\sqrt{x^2+10^2})}{\sqrt{x^2+10^2}}=0$

$2x=\sqrt{x^2+10^2}$

$4x^2=x^2+10^2$

$3x^2=10^2$

$x^2=\frac{10^2}{3}$

$x=\frac{10}{\sqrt{3}}$

The first derivative test shows we have a minimum at this point, so the minimum cost is:

$C\left(\frac{10}{\sqrt{3}} \right)=950\sqrt{3}+1520\approx3165.44827$

Thus, the minimum cost is about $3,165,448.27.

**MarkFL** · October 29th 2012, 01:36 AM

Since you posted in the algebra forum, I assume this is not homework, but you are just curious how it is to be solved. You are very close with your trial and error estimate, by the way, although your computed cost is wrong, but your value for x is very close.

This is a problem typically given to Calc I students.

**bradscoolio** · October 29th 2012, 01:50 AM

Cheers! The cost I typed in was just a typo and not actually what I intended :P. The work set by my teacher was only to calculate 10 different lengths but i was too curious and wanted to know how it was done. Thank you very much, well appreciated!
ps: I'm only in year 9 at highschool and there is no way in hell that my teacher could expect me to work out the cheapest rout :P

**cac2008** · October 29th 2012, 02:31 AM

I'm glad someone responded with a solution using algebra - I got horribly confused around the differentiation (I need to go back over my old notes methinks!) and solved it instead using a Monte Carlo. It felt a bit like using a sledgehammer to open a peanut but it came close to the calculated value, although i'm not sure why the estimated value is more around 3.80 as opposed to 3.17

what is the cheapest way to lay the pipe?-cost-function.jpg

AK

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