The C&M Express Company is setting up some warehouses to make deliveries around the Chicago area. To fit the layout of the roads, they chop Chicagoland into identical squares each of area x . A warehouse will be built at the center of each square, and deliveries destined for a location within a square will be made from the warehouse located within the same square. The engineers tell us that to cover each square from a warehouse located at the center, the transportation costs per item are proportional to Sqrt[x] and the warehouse cost per item is proportional to 1/x .

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Question

The C&M Express Company is setting up some warehouses to make deliveries around the Chicago area. To fit the layout of the roads, they chop Chicagoland into identical squares each of area x . A warehouse will be built at the center of each square, and deliveries destined for a location within a square will be made from the warehouse located within the same square. The engineers tell us that to cover each square from a warehouse located at the center, the transportation costs per item are proportional to Sqrt[x] and the warehouse cost per item is proportional to 1/x .

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anonymous · Answer

So the overall cost of servicing a square is
      $$a \sqrt{x}+ \frac{b}{x}$$
where a and b are positive constants.

How should you set x in terms of a and b to make the overall cost of servicing a square as small as can be?

anonymous · Answer

and I guess to start, if I knew the meaning of 'set x in terms of a and b' I might have a shot at working this out... does that mean the solution will look like a= ? , b=? or I find a value for x? and any idea what a and b actually represent in real world terms?, does it even matter?

dan815 · Answer

a and b are arbitrary proportionality constants

dan815 · Answer

$$cost(x)=a\sqrt {x} +\frac{b}{x}\
cost'(x)=?$$

dan815 · Answer

you can set the first derivative =0 to see if there is a minimum, and check the endpoints

anonymous · Answer

awesome dan, I actually had a go at this, plotted it out and took the derivative to find the dip of the trough... is the solution then that the lowest point of the plot would be the most economical point for x ?

dan815 · Answer

u can rewrite x in terms of a and b from setting the first derivative = 0

dan815 · Answer

the solution is simply a function of the arbitrary constants a and b,

anonymous · Answer

damn, I have to go right now though.. being called to dinner...back in an hour.. can I hit you up then?

dan815 · Answer

im about to go to sleep :)

anonymous · Answer

thank you by the way.

dan815 · Answer

sure thing, someone els will come and help you though, @Luigi0210

anonymous · Answer

so if anyone is looking ... is this what you mean?

$$ a -> \frac{(2*b)}{x^{\frac{3}{2}}}$$

and

$$ b -> \frac{(a x^{\frac{3}{2}})}{2} $$

anonymous · Answer

I took this function...
$$ f[x] = \frac{b}{x} + a* \sqrt{x}$$
then got the derivative as.. 
$$ f'[x] = -\frac{b}{x^{2}} + \frac{a}{2\sqrt{x}}​ $$
then rearranged to get...
$$ a -> \frac{(2*b)}{x^{\frac{3}{2}}} $$
and
$$ b -> \frac{(a x^{\frac{3}{2}})}{2}$$

anonymous · Answer

@Hero, @preetha hi guys, could you give this a quick look and let me know if I'm on the right track.. appreciated, thanks.

anonymous · Answer

greets @hartnn , sorry to ambush you, can I hassle you for a sec to check on my work over here?

hartnn · Answer

they are asking for x as a function of a and b 
 x= f(a,b)

which means, we need to isolate x from 
$\Large f'[x] = -\dfrac{b}{x^{2}} + \dfrac{a}{2\sqrt{x}}​ = 0 $

hartnn · Answer

"How should you set x in terms of a and b"
right? 
so they are expecting,
x = ... some terms in a and b ...

hartnn · Answer

first step 
$\Large-\dfrac{b}{x^{2}} + \dfrac{a}{2\sqrt{x}}​ = 0 \\Large \dfrac{b}{x^{2}} = \dfrac{a}{2\sqrt{x}} $

try to go ahead.. :)

anonymous · Answer

I get 3 solutions ....
Here's one of them.

$$ x -> \frac{
 (-2)^{\frac{2}{3}}*b^{\frac{2}{3}}}{a^{\frac{2}{3}}} $$

anonymous · Answer

The other two are imaginary, so I think I can ignore them

hartnn · Answer

thats an complex number right?

chose a real number from the 3 solutions

hartnn · Answer

(-2)^(2/3) is complex number

hartnn · Answer

2^(2/3) is a real number :)

anonymous · Answer

If I multiply that out.. will it be considered a real number?

anonymous · Answer

$$ x -> \frac{1.5874010519681996*b^{\frac{2}{3}}}{a^{\frac{2}{3}}} $$

anonymous · Answer

lol, I know I ask the dumbest questions

hartnn · Answer

imaginary/complex numbers come into picture when we have a negative number inside a square root ... 

a,b are positive, so forget them,
(-2)^(2/3) does have a negative number
same for $\sqrt[3]{(-1)}$

so the only real solution is what you posted above :)

though 2^(2/3) looks better, keep it that way

and the only dumb questions are the questions not asked!

hartnn · Answer

and 
$\Large x  = (\dfrac{2b}{a})^{\dfrac{2}{3}}$
looks even better...

but what looks better is subjective ;)

anonymous · Answer

ah I like that.. I had a lot of 2/3's .. was just thinking they probably factored out..

anonymous · Answer

once again.. you rocked it .. thank you

hartnn · Answer

most welcome ^_^
always happy to help :D