Looking for assistance with a math problem.

[magnumopus]

Blessings to all! I am having trouble with configuration of a math equation to represent a real life scenario for my creation of my dream home. I have difficulties with math, mostly in the sense of lack of experience, math actually fascinates me, however in my younger years I didn't develop a good foundation and it has been trying to add to the lack of stability. My wife and I are in the beginning stages of creating our first dream home. We believe in renewable energy and are considering a home designed as a pyramid, for multiple purposes. The problem I have first come across in developing my initial plan, to give to an architect is with the scale and size of my home. I am trying to get a baseline to figure out the area needed to make a home that has a minimum of what we need. So, here is my feeble attempt at giving a word problem to explain.

The design is a 4 floor pyramid (above ground, there will be a basement below but it will be squared), the top tier/ floor needs to have a useable area of 10 feet by 10 feet square that is at least 7 feet tall. With that said the overall area of the floor for the 4th tier will be larger than 10 x 10 and the middle apex of the ceiling will be higher than 7 feet as its a pyramid. So it will be a 10x10x7 room inside a small pyramid which is the top floor to the overall pyramid. Im trying to figure out how to create a math equation with the information I desire to produce the size of the other tiers if I wish them to be 7 foot tall ceilings in the usable area, as well how big will the bottom floor of my base need to be with that top tier being what i need (10x10x7). As well I am trying to configure at what degree of slope this will be. I am trying to see if what I want is possible without getting to massive. I fear that with a useable area that big in the top floor/ tier that the bottom floor is going to be too big and the slope to relaxed.

it seems like simple math, but I don't know where to start?

 

[Ishuda]

If I am understanding your problem correctly, there is no unique pyramid other than a basic, 'it must be higher than 28 feet' and have a base greater than 10' square. The way I came to this conclusion was to envision a cross sectional view of the pyramid for the top floor through the middle of the pyramid. That would give an isosceles triangle [two sides are equal] and we could take the bottom of the triangle to be equal to a height of zero (the x axis) and the left and right sides as the face of the pyramid. Now put a point 5 inches (scale 1 inch equal 1 foot, so representing 5 feet) to the right (point at (5,0)), then one 5 inches to the right and 7 inches up (point (5,7)). The line formed by that would represent the wall of the room. Now draw any line going through (5,7) and crossing the x axis to the right of (5,0). As the slope of the line representing the face of the pyramid gets smaller in magnitude the square foot of the floors below get less and the height of the pyramid gets higher. Conversly, as the slope get larger the size of the rooms would get larger.

So, to nail down the actual pyramid size, you will need to also chose the square footage of a room below the top floor.

BTW: The solution can be obtained by looking at the angle of the line representing the face of the pyramid and realizing that the tangent can be represented by different ratios.

Edit: Sorry, got those slopes mixed up (turned around) as Quaid pointed out below.

 

[Ishuda]

Oh, and if you really want to get fancy, you can write the solution as a function of the 3rd floor size (for example) and then minimize the area of the resulting pyramid

 

[Quaid]

what does that mean?
was assuming you wanted exactly 10 by 10...

It means that he wants a 10 by 10 by 7 foot space stuffed inside the top of a pyramid. (The OP states that the fourth floor must be larger than 10 by 10, and that the center of the ceiling will be greater than 7 feet.)

I agree with Ishuda; there are many different pyramid dimensions to accommodate such a fourth floor.

One could build a pyramid with steeply-sloped sides; the tip of the pyramid would extend much farther than 7 feet above the fourth floor.

One could build a pyramid with not-so-steeply sloped sides; the tip of the pyramid would extend not much farther than 7 feet above the fourth floor.

The former pyramid would have a relatively small footprint. The latter pyramid would have a relatively large footprint.

We need more information, to determine the either the slope of the outside walls OR the dimensions of the footprint. (We can ignore the basement.)

 

[magnumopus]

Wonderful!!! Clarity!?!

In immediate response, I was probably poor at explaining appropriately. I feel that some of you are tracking with me. I had wondered if you would need more information to create the equation. To clarify, well attempt to clarify, the top floor, the "dog house" is a meditation room, it would ideally be a room for all of our family to "rise above" and just simply be. We are Christians so I would hope it to be a room for one on one with God, but I would enjoy if any of our friends needed to come and just decompress/ pray to whatever God or Being they chose. In either case, I figured if I wanted each floor to be around 7 feet and the top room/tier to have a useable area of 10 by 10 by 7 that one of you genius's could figure out the base and slope. I tried making a drawing and transferring it but, was unsuccessful. So picture a triangle, within that triangle is 4 rectangles, each one representing my needed living space area. Of course I will have to take into consideration structural necessities, but before I can even get there I am trying to see what kind of space I am looking at. I don't want a massive area on the bottom, but I want the top floor to be usable, so thats why I started by figuring it, my smallest space. I asked my wife what kind of area would we need for all forms of relaxation, weather it be sitting and reading scripture, yoga, a few people doing aerobics or dancing, anything, whats a good size for area. A 10 by 10 foot room that is 7 feet tall, seems small but because its inside a pyramid, the floor will actually be larger than 10 feet and the ceiling will be taller than 7 feet. Remember the vision, triangle with 4 rectangles inside, I'm referring to the top rectangle, I need that rectangle to be 10 by 10 by 7, but if you take away those lines and leave the 3 other rectangles the area left is actually what the top tier will be. Im sorry I can't put this into better terms, I had some friends in college that brainstormed like you all did, I was an exercise science/ english major and math isn't my thing. I also ignorantly assumed that if one could figure out the one tier and keep each floor 7 feet tall that it would determine the slope of the pyramid. I don't want a massive base, so I was hoping once an equation was developed I could add or subtract to change the areas as needed. See if you wonderful people can consider this, I thank you so much for your insight thus far, I am going to jot it all down and start trying to get "mathletic" with it. On a separate note, I think this is great, people getting together, helping each other with their intellect, its wonderful. Many Blessings, and I thank you.