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Question 577418: Hello,
I am currently working on a Math project, and am on the final instruction of the project. I have had to find design a building structure, with office blocks (2.5 m in height) on the building structure (so each floor is 2.5 m in height). Now the final task is to find the area of the volume wasted space of the structure (shown in the link below, the wasted space are the triagles but the office blocks).
Now I'd know how to calculate the volume, but the only problem I have is finding the length of each office floor (since each one is different). I have asked my math teacher and he has told me there's an equation to work out the length of each individual floor. I have included a second image of just a single floor if this in any way helps.
Thank you very much :)
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! The only problem is, we do not know what function f(x) defines the outer (curved) edge of the building, so we cannot find an exact value. Additionally, this function is not one-to-one so we would have to partition this function to have an inverse function.
Suppose f(x) is defined only on [0,75]. The cross-sectional area of the building bounded by x=0, x=75, y=112.5 is equal to
dx)
We can write this integral in terms of dy, but we don't have to since it's a definite integral and we will still get the same result.
The cross-sectional area of each floor is given by )) . We find this by subtracting off area from a rectangle of size 2.5 * 75. Sum this from x = 1 to x = f, where f is the last floor counted:
))
The error E is taken by finding A - A1:
dx - \sum_{i = 1}^{f} 2.5(75 - f^{-1}(2.5x)))
However you want to multiply this by 2, because we only accounted for half of the building. Therefore the area of what you want is 2E. To find the volume, multiply by the width of the building (72 m).
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