Expert: Josh - 10/4/2008

Question
Hello
I am s secondary school student and I need help with this question at least some ideas about how to solve.
Thank you
Two villages, Abu and Bip, lie at distances 2km and 1km above a river. River is 4km long and Abu is at start of the river bib is at the end. It has been decided to build a fresh water supply station along the river to supply both villages. Where along the river should the station be located in order that the total pipe length be kept to a minimum?


Answer
Hi Helena,

Thanks for letting me know you are a secondary school student. I can now answer your question to better suit your needs.

I think we can describe the situation with a picture. Note: To line things up and give you a proper perspective, I need to draw it sideways.

X-----------A
|
|
|
Y
|
|
|
|
Z-----B

Here, XZ represents the length of the river. Town A is at an elevation of 2000 m above X while town B is at an elevation of 1000 m above Z. To get fresh water supply, the station must be built at sea level, somewhere between X and Z. We let its position be Y in the diagram.

Your task is the find length w=|XY|, such that the distance |AY|+|YB| is minimized. The constraints are as follows:
|XZ|=4 (kilometer)
|XZ|-|YZ|=|XY|=w
0<=|XY|<=4

To solve this, write an expression for distance |AY| and |YB| in terms of the unknown variable |XY|=w.

|AY|=sqrt(|AX|^2+|XY|^2)=sqrt(4+w^2)
|BY|=sqrt(|BZ|^2+|ZY|^2)=sqrt(1+(4-w)^2)

Let total pipe length be D=|AY|+|BY|.

You apply standard techniques (i.e., differentiate D, set the derivative dD/dw to 0 and solve for w)

Different perspectives:

(1) We can define cost in different ways. An unconventional cost measure is R=|AY|^2+|BY|^2. In this case, the algebra is greatly simplified without the square root parts.

R=4+w^2+(17-8w+w^2)
=2w^2-8w+21
dR/dw = 4w-8 = 0 has solution w=2 (Interestingly, the common man is not necessarily wrong in suggesting a station half way between X and Z, it just depends on how the cost function is defined. The way it is defined here, they are absolutely spot on, it actually minimizes cost although it would mean that the cost of construction is not directly proportional to the pipe distance).

(2) The "proper" way is to define D=|AY|+|BY|.

D=sqrt(4+w^2)+sqrt(17-8w+w^2) OR
D=(4+w^2)^(1/2)+sqrt(17-8w+w^2)^(1/2)

Going forward is a bit of a mess. I'm just going through the motion at this point, so watch out for errors. You should definitely do/check this yourself.

Set dD/dw = (1/2)*(2w)/sqrt(4+w^2)+(1/2)(2w-8)/sqrt(17-8w+w^2) = 0
w*sqrt(17-8w+w^2)=(4-w)*sqrt(4+w^2)
w^2*(17-8w+w^2)=(16-8w+w^2)(4+w^2) ...next, square both sides
17w^2-8w^3+w^4=64+16w^2-32w-8w^3+4w^2+w^4 ...cancel like terms
0=64+3w^2-32w

possible solutions:
w = [32+sqrt(32^2-4*3*64)]/(2*3), [32-sqrt(32^2-4*3*64)]/(2*3)
= 8, 2.666...
legitimate solution is 2.666 (barring errors)