 # Curved Path, Trapezoid Cross-Section

The following e-mail was sent to me:

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I am an engineering geologist. I am involved in a case where a landowner filled in a stream with a road, several bridges, and an island. The local regulatory agency would like to estimate the volume of fill placed in the stream. That's my job. I have been able to calculate most of the volume by slicing the fill into cross sections and calculating the area of cross section and then multiplying the area for whatever length I decide to evaluate. My problem is where some cross sections are attached at a pivot point and I have a angular "length" to evaluate. For example how would you calculate the volume where two trapezoid cross sections of different areas that have a common pivot point and are separated by 50 degrees? Or more easily, what is the volume of a trapezoid that sweeps across an angle of 50 degrees?

No wonder I am a Geologist, couldn't handle the mathematics! Thanks for your help.

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Interpret what he is asking for and devise an easily understood strategy for estimating the volume he is looking for.

There is a pivot point P and a distance R from the pivot point to the center of the trench. We have a trapezoid cross section with bases  a  and  b and height  h. The pivot angle is 50 degrees.

Click here for a specific Example where  a = 20 ft.,  b = 10 ft.,  h = 12 ft. , and   R  = 100 ft.  is the radius for the sweep of 50 degrees.

Devise a spread sheet that can be used to estimate the volume given input of the dimensions of the trapezoid, the distance from the pivot point to the center of the trench, and the pivot angle.

Write a draft of what you might send out mathophobic geologist as your response to his e-mail.

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