Problem

The Slope of a Pyramid:

How steep are the sides of the pyramids of Egypt? The use of mathematics allows us to measure steepness. One measure of steepness is the slope.

1. Use cubit measures to find the slope of the triangular face of the pyramid.
2. Repeat the process using feet.
3. What do you notice about the slope measured in these two ways?

You may wish to include your calculations, and explanation in your portfolio.

Answer

My answer to the question is that the pyramid has a slope in: Cubits: 25/18

Feet: 143/103

-Here is how I solved the problem-

For all the following, I used the front face of the pyramid.

1. To solve this problem, I first had to find two points.

My first point was (0,250)

I found this point by:

Making a grid shown to your right

The triangle (front face) goes up 250 cubits, but stays 0 on the X-axis.

Therefore the point is (0,250)

 

 

My second point was (180,0)

I found this point by:

Making a grid shown to your right

The triangle has a base of 360 cubits

From the origin to the corner is half-way (180 cubits)

The point is still 0 on the y-axis,

Therefore the point is (180,0)

 

After I had my two points, I wrote out the equation to get the slope:

Rise = Delta Y = 250 - 0 = 250 =

Run = Delta X = 0 - 180 = 180 = 25/18 cubits

Delta Y is the difference in the two, y #’s from the two points,

So, I subtracted 0 from 250

Then, I had to subtract the y and x in the same order,

So, I subtracted 180 from 0

From their, I got 250/180

Then, I divided the fraction by 10, to reduce it to 25/18 cubits

Leaving me with an answer of 25/48 cubits

 

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2. To solve this problem, I used the same method as before.

I first found my two points.

I found my first point (0,429), by:

Making a grid (to your right), and placing the triangle onto it

The triangle (front face) goes up 429 feet,

And stays 0 on the x-axis,

Therefore the first point is (0,429)

I found my second point (309,0), by:

Making the grid shown below

The triangle goes out from the origin 309 feet

It stays 0 on the y-axis,

Therefore the second point is (309,0)

 

 

With the two points, I was able to do the same equation as with cubits:

Rise = Delta Y = 429-0 = 429 =

Run = Delta X = 0-309 = -309 = -1 40/103 = -143/103 cubits

Delta Y is the difference in the two, y #’s from the two points,

So, I subtracted 0 from 429

Then, I had to subtract the y and x in the same order,

So, I subtracted 309 from 0

From their, I got -429/309

Then, I reduced the fraction to -1 40/103 cubits

The next thing I did was to change my mixed # fraction into an improper fraction (143/103)

Leaving me with an answer of -143/103 cubits

 

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3. I noticed three things about the slope measured in these two ways.

a) I figured out the ratio of cubits to feet in each pair of numbers, and there is a .34 difference in each.

 

2. The cubits that Egyptians used to measure were larger than feet.

 

3. The slope measured in feet is very close to the slope measured in cubits.

Feet: 1.388349515

Cubits: 1.388888889