Maximizing Flow in a Rain Gutter: 3-sided vs Semicircular Cross Section

In summary, the metal must be bent into a symmetric form with three straight sides in order to create a rain gutter. The cross section is shown below. It would be better to create the gutter with a semicircular cross section, but without a constraint, the Lagrange multiplier method cannot be used. The equations for the cross sectional area are given, and the maximum is found to be when b is the leg of each of the right triangles in the corners.
  • #1
clairaut
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A long piece of sheet metal w inches wide is to be bent into a SYMMETRIC form with three straight sides to make a rain gutter. A cross section is shown below.

\_____/

The base is w-2x and the angled side lengths are both x with a theta between top horizontal.

A. Determine dimensions that allow maximum possible flow or maximum cross sectional area.

B. Would it be better to bend the material into a gutter with a semicircular cross section than a three sided cross section?
 
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  • #2
I can't seem to find a constraint. Without constraint, I cannot use lagrangian method.

So, I take the partials of the cross sectional area function with respect to x and then with respect to theta.

I get nowhere afterwards.
 
  • #3
First this is clearly a home-work type question and should have been posted there.

Second the "constraint" is x= w/4.

Frankly I wouldn't consider this a "Lagrange multiplier" problem to begin with. I would write the volume of the trapezoid as a function of x and [itex]\theta[/itex], replace x by w/4, and treat it as a "Calculus" I problem of finding the value of the single variable [itex]\theta[/itex] that maximizes the area.
 
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  • #4
Why would constraint be w/4
 
  • #5
Also, volume of trapezoid is not important here. I only need the cross sectional area of this trapezoid.
 
  • #6
?
 
  • #7
Show some working out please. Give us the formula for the cross-sectional area, it'll have 3 variables: w, x, Θ. If you can do that, we can start to help.
 
  • #8
[A(x,b)] = (x^2-b^2)^.5 (w + b - 2x)

Where b is the leg of each of the right triangles in the corners. [A(x,@)] = (w)(x)[sin(@)] + (x^2)[sin(@)][cos(@)] - 2x^2[sin(@)]

Above are the cross sectional surface area equations.
 
  • #9
clairaut said:
[A(x,b)] = (x^2-b^2)^.5 (w + b - 2x)

Where b is the leg of each of the right triangles in the corners. [A(x,@)] = (w)(x)[sin(@)] + (x^2)[sin(@)][cos(@)] - 2x^2[sin(@)]

Above are the cross sectional surface area equations.

I get the same formula although you should try to simplify it a little. Okay, you want to find the maximum. Can you see how to do it? It will of course have something to do with the gradient. Refer to your textbook if you need to.
 
  • #10
Yes. No problems with gradient. It's the CRITICAL POINTS that are hard to find.
 
  • #11
I've helped all I can, sorry.
 
  • #12
Lol. How have you helped?

You've merely agreed with me... No more
 
  • #13
clairaut said:
It's the CRITICAL POINTS that are hard to find.

Please post an attempt. Also, this thread belongs in homework, so please post there in the future.
 

What is the purpose of this experiment?

The purpose of this experiment is to determine which cross-sectional shape, between a 3-sided and a semicircular shape, maximizes the flow of water in a rain gutter. This can help in designing more efficient rain gutter systems for buildings and homes.

How is the experiment conducted?

The experiment involves creating two rain gutters, one with a 3-sided cross section and the other with a semicircular cross section. The gutters are then filled with a set amount of water and the flow rate of water through each gutter is measured and compared.

What factors affect the flow rate in a rain gutter?

The flow rate in a rain gutter can be affected by several factors, including the cross-sectional shape, the angle of the gutter, the material of the gutter, and any obstructions or debris in the gutter.

What are the expected results of the experiment?

Based on previous studies and theories, it is expected that the semicircular cross section will have a higher flow rate compared to the 3-sided cross section. This is because the semicircular shape has a smoother surface and allows for more efficient flow of water.

How can the results of this experiment be applied in real-life situations?

The results of this experiment can be applied in the design and construction of rain gutter systems for buildings and homes. By using the cross-sectional shape with the highest flow rate, the efficiency of the gutter system can be improved, thereby reducing the risk of water damage and improving the overall functionality of the system.

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