



 
Steven sent us this diagram Hi Steven. Thanks for sending us the diagram  that clarifies things quite a bit. I note that the radius in your diagram is 240'6" whereas in your earlier message you said 187'6", so I presume the diagram (240'6") has the correct value. This is a lot like a problem I solved in 2007 here: http://mathcentral.uregina.ca/QQ/database/QQ.09.06/s/angela1.html On that web page, I showed how to derive a general solution. We can use the results of that work to speed us to our goal. First though, I should notify you that the curvature is very slight: a chord of length 34'8" is tiny compared to the radius of 240'6", so the maximum height of the arch is going be be very small as well  if this isn't at all what you were expecting, then review whether your radius is really that large. Okay, to begin, we calculate the apothem using Pythagoras. That is, if you draw a line connecting the midpoint of the chord (the flat base of the arch) to the center of the circle and connect the endpoint of the chord to the center as well, then you have constructed a right angle triangle. We know the hypotenuse is the radius (240.5 feet) and the short side is half the chord length, which makes 17.33333 feet. The long leg is the apothem (I'll call it A) and can be found using Pythagoras: A = √(240.52^{2}  17.33333^{2} ) = 239.8746 feet. Thus for my formulae on the other web page, we have x_{0} = 17.33333 feet, y_{0} is 239.8746 feet and r^{2} = 240.5^{2} = 57840.25 square feet. I'll convert from feet to inches, so now x_{0} = 208 inches, y_{0} is 2878.5 inches and r^{2} = 8328996 square inches. Okay, so now we can compute the heights at 16 inch intervals starting at one end. Thus, starting from one column and moving away at 16 inch intervals (the x coordinate), we have the following heights (y coordinate) in inches:
As you can see, the height at the columns is 0 inches (as expected) and the height in the center is 7.5 inches. That's a very slight arch, but hopefully it is what you are seeking. Cheers,
 


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