Volume calculation between two different trapezoidal areas

[americanfrank]

I am designing a trapezoidal prism basin that must contain a certain volume per acres contributing to the basin. Therefore, I would like to create a function that will size the basin.

My question is: how can I determine the volume between two different areas? In this case these two areas are trapezoids. I know A = a((b1+b2)/2) for area of a trapezoid.

Let's say b1 is the smaller side of the trapezoid.

In this case trapezoid 1 and 2 will have b11 = b12 while a1 and a2 will not be equal and neither will b21 and b22.

I believe that knowing this information will be the key to determining other partial volumes that I will ultimately sum to provide the total volume.

Thank you for any help you may have!

 

[LCKurtz]

Here's what I would try. I will change the notation a bit for clarity (for me at least). Let's call the lower base of the two trapezoids b₁ and b₂, the upper bases B₁ and B₂,the heights h₁ and h₂, and the distance between these two ends L. If I understand your question correctly you want the area of a parallel trapezoidal cross-section at some distance x between 0 and L.

The upper base will vary linearly in x from B₁ to B₂, so its equation would be, in terms of x:

B(x) = B₁(1 - x/L) + B₂(x/L)

Notice that B(0) = B₁ and B(L) = B₂ and this is linear in x.

You can do the same thing with the lower base and height to get formulas at x. Then use your area formula with these functions of x.

 

[americanfrank]

Thank you for your response. However, I am confused. When you say "area of a parallel trapezoidal cross-section", do you mean volume between two parallel areas?

Thanks.

 

[Studiot]

What you are seeking is called the prismoidal formula by surveyors and civil engineers.

It is used in earthworks calculations.

http://www.google.co.uk/#sclient=ps...gc.r_pw.&fp=658d444f97a7fc20&biw=1024&bih=585

 

[LCKurtz]

Thank you for your response. However, I am confused. When you say "area of a parallel trapezoidal cross-section", do you mean volume between two parallel areas?

Thanks.

No, I mean the area. You have to integrate A(x) from 0 to L to get the volume. Here's a picture: (the sick looking greenish color represents the bottom of the prism.)

Of course, I may have interpreted your problem all wrong, in which case you should have included a picture in the first place.

 

[americanfrank]

Thanks LCKrutz. This is what I needed to know.

 

[LCKurtz]

Is this homework? If not, do you want to know the volume of that to check your work or would you rather do it yourself?

 

[americanfrank]

no, its an actual application. Designing unique sediment traps for a VDOT job. I needed to determine the storage capacity based on the contributing area from a storm event.
I am trying to create an excel spreadsheet that will output usable information in each sediment trap's analysis.

sounds like I'll need to use some integration...which I need to brush up on.

 

[LCKurtz]

no, its an actual application. Designing unique sediment traps for a VDOT job. I needed to determine the storage capacity based on the contributing area from a storm event.
I am trying to create an excel spreadsheet that will output usable information in each sediment trap's analysis.

sounds like I'll need to use some integration...which I need to brush up on.

OK. In that case, for your reference, here's the answer to the volume of the figure I drew.

V = (L/12)*(2*h₁*b₁+h₁*b₂+2*h₁*B₁+h₁*B₂+h₂*b₁+2*h₂*b₂+h₂*B₁+2*h₂*B₂)

 

[Studiot]

What was wrong with the prismoidal formula?

It is exact for your application.