Optimization of a suspended system. (hanging mass)

[stf]

Homework Statement

A load must be suspended 6m below a high ceiling using cables attached to two supports that are 2m apart. How fare below the ceiling (x in figure) should the cables be joined to minimize the total length of the cable used? They give a figure, which I am butchering here.

  |2m|
--------------------
   \  /      |
    \/       |x is this length
    |
    |
    |        | 6m is from top to bottom
    []

Homework Equations

This is in Calc 1, under optimization problems. So generally we are looking at minimum and maximum values using first and second derivatives.

The Attempt at a Solution

This is clearly a minimization problem, which means I will be taking a first derivative at some point. I am perplexed as to what actual form to put this in though, as I know I need to ultimately find the length of cable, which would seem to be something like 6 + x, but this does not feel right, and from here I don't know where to go.

 

[tiny-tim]

hi stf!

with questions like this, you need to choose a variable to work with

you can choose anything reasonable, but i'd suggest using the length of the vertical part …

what do you get? ​

 

[stf]

hi stf!

with questions like this, you need to choose a variable to work with

you can choose anything reasonable, but i'd suggest using the length of the vertical part …

what do you get? ​

But how would I go about setting that up in an equation? It seems like I would have two parts of the length of the wire, the two that form the intersecting \/ so for example 2a, and the rest to the bottom, or (6 - a), but if i set this up to be a whole length of cable it would be (6 -a) + 2a which is just 6+a, but this doesn't seem to lead anywhere.

 

[tiny-tim]

if the vertical part is x, the other parts will be longer than 6 - x

 

[stf]

Well, I mean it seems like it would have 3 parts, two identical which are the \/ section and one long remaining section. the total from top to bottom would be 6m, so the vertical would be a portion of the 6m minus the height in between the connecting angle, so i don't really understand what you are getting at here. I understand that the two parts will be longer than a simple straight line due to the triangle formed, but I don't really understand how to proceed.

 

[tiny-tim]

pythagoras

 

[lucy8love]

But how would I go about setting that up in an equation? It seems like I would have two parts of the length of the wire, the two that form the intersecting \/ so for example 2a, and the rest to the bottom, or (6 - a), but if i set this up to be a whole length of cable it would be (6 -a) + 2a which is just 6+a, but this doesn't seem to lead anywhere.

I'm actually working on the same problem...I don't know if you got it yet, but I've set it up pretty close to that, with one difference:

Instead of 2a + 6 - a = L (L being total length), I have

2a+6-x = L

(x being the distance between the top and the 'fork').

Does that make sense to you? From there, I'm going use Pythagoras to get a in terms of x, and use that to finish the problem.

If I've made a grave error, please correct me. :)

EDIT: Having done all this, I seem to have gotten a reasonably sensible answer--note, once you have a in terms of x, you can substitute the the equation into a, in the length equation so you have that equation all in terms of x and can properly differentiate it.

 

[stf]

I'm actually working on the same problem...I don't know if you got it yet, but I've set it up pretty close to that, with one difference:

Instead of 2a + 6 - a = L (L being total length), I have

2a+6-x = L

(x being the distance between the top and the 'fork').

Does that make sense to you? From there, I'm going use Pythagoras to get a in terms of x, and use that to finish the problem.

If I've made a grave error, please correct me. :)

EDIT: Having done all this, I seem to have gotten a reasonably sensible answer--note, once you have a in terms of x, you can substitute the the equation into a, in the length equation so you have that equation all in terms of x and can properly differentiate it.

Hello! Actually I am a terrible person, I solved it after tim's last post and should really have thanked him.

So yes, I eventually set up the equation to be (6-x) + 2(sqrt(1^2 + x^2)) where x is the length of the "triangle" that is formed by the two cables connected to the beam, which is exactly what you are doing I think.

 

[lucy8love]

^^Haha yes, quite literally I did the exact same thing.