Mathematical modeling - a task

[psihomehanicar]

Can anyone help me with this asignement? I think im shouldn't be a problem to those who are good at maths, and it would mean a lot to me.

Here it is:

Do a project of a bussiness block in the building that has shape like in this picture:

 

http://img163.images...i/imagewm0.jpg/

 

Building has a rectangular base, dimensions: 150m length, 72m width.Maximal height of the structure must not be higher than 75% of its width because of the

stability of construction or less than half of its width because esthetic reasons.Minimal height of room in public buildings is 2,5m.

 

1) Form a model of structure with curved roof(shown on the top picture) in case that height of building is 36m

2) Measure dimensions of cuboid with maximal volume that could fit in the structure with oval roof.

3) Using technology(Wolfram Mathematic, Sage Math, MatLab, or some of the programs that simulate work of the graphic calculators like Dream Calc...) examing

how changing dimensions of this structure affects on changing dimensions of the biggest cuboid that can fit in that structure.

4) For different values of structures height(examined in last step) determine used space ratio.

5) Calculate maximal surface of business block for different values of height, within given specifications.

 

Buildings heights are 40, 42, 44m.

 

I hope you will be able to help me to do at least one part if not whole asignement.

[the tree]

The thing is, there's kind of a very wide of curves that could define the shape of the roof, it could be quadratic or a sine wave, my guess would be a catenary. I certain amount of trial and error is probably needed to get that to work how you want it.

 

Edit: Right yes, just played with Maple for a while, this isn't too hard but the formulae are a bit ugly.

 

Let's say we have parameters l,w,h for the desired length, width and height.

1. You should be able to put together an equation u(x,y) for the height of the building at a given point. Too much help here would really jeopardise your work, so I wont give you further hints until you say what you've done. Mine was based around cosh but if you do something entirely different then please do tell.
   I'm assuming you've done the above so that u(x,0) is the interesting cross-section, and u(0,y) is a rectangle of size l*h. And that the centre of the floor is at (0,0).
   
2. Then some quick sketching should show you that the volume of the cuboid with sides at x and -x can be given by l*u(x,0)*|x|.
   Plot, differentiate, find the stationary point etcetera.
   
3. Play aimlessly with software of your choosing, make pretty pictures, have fun.
   
4. I'm sure you can integrate to find the area under the curve, do that for different h. Repeat part 2 for each of those. Compare the ratios.
   
5. Check your notes for an arc length formulae.
   

Edited December 1, 2010 by the tree