Pyramid step height question: is my math correct? [closed]

Question

I'm writing a bit involving a pyramid. The pyramid is 400 feet in width, 150 feet tall, and one side has 100 steps ascending to the top. The first step is 36 inches (3 feet) tall, and each successive step is 5% shorter in height than the step before it. How tall is the 50th pyramid step, and how tall is the 100th (last) pyramid step?

I used the following math: To get the length of the diagonal, I used the Pythagorean Theorem: a^2 + b^2 = c^2. I used half the width of the pyramid in order to form the right angle required for the theorem (200 feet).

200^2 + 150^2 = 62500.

The square root of 62500 is 250, so the diagonal is 250 feet long.

In order to compute the height of the last step, I thought about adapting the formula for compound interest, and substituting a negative interest rate (-.05).

This equation looks like:

50th step height: 36(1+.-.05/1)^50 = 13.11 inches

100th step height: 36(1+-05/1)^100 = 0.21 inches

Is this math correct?

Answer 1

Spreadsheet time!

I made quickly a spreadsheet to simulate this degrading sequence¹. This has the benefit to need less math knowledge to build and understand :). Here are the first rows of the table.

And then the 50th, like you asked...

And finally the last steps!

As you can see, we've got some problems.

• First, you don't reach the 150 feet you'd need to reach the top, quite far from it.
• Then, your last steps are only 0.02 feet high². It's a quarter of an inch, or 0.61 cm. While I know that pyramids are too well-made to not be built by your aliens, any kind of long-term tear will fuse your last steps together. See how the Great Pyramids are feeling old and tired. Unless there's no erosion or other weardown, your mathematically perfect pyramid won't be... So mathematically perfect in a few centuries 😓.

How to improve it?

With the spreadsheet (steps to reproduce below¹) you can already make quick tries to reach the height you want, without needing to calculate sequence/function limits. If you're unsure what to tweak but want to keep your formula, you can :

• Increase the initial value. Note this will scale uniformly the size of your pyramid (if you choose 6 instead of 3, your pyramid's final height will be twice)
• Decrease the size loss on each step (e.g. : from 95% to 97%).

Since you're making a lot of steps, your best bet would be to reduce the size loss. Indeed, the last step size is basically your first step size multiplied by 95% almost one hundred times!

Height(100) = Height(99) * 0.95 = Height(98) * 0.95² = ... = Height(1) * 0.95⁹⁹ -> multiplied by ~0.006!

To keep the last steps tall enough, you'll probably need to reduce the overall number of steps or increase the pyramid's size. It'll be quite complicated to make visible steps with one hundred for 150 ft.

What about length?

Now that you found your steps height, it's time to calculate the steps length so that it's proportionate mathemagically. Since upscaling and downscaling the steps height/length gives proportional results regardless of your decreasing factor, use it to calculate the initial and subsequent steps length :).

E.g. : Let's say you only upscaled the initial step height to 7.5 ft (it gives roughly 150 ft for 100 steps). Your pyramid is 150 feet tall and 200 feet wide from the center, so to get the steps length, multiply the steps height by 200/150 = 4/3 = ~1.33.

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^{1 : To reproduce it in Excel, input the initial value 3 at A1 (or A2 with a header). Then, for the height of the second step, input at A2 : "=\$A1 * 0.95". Drag down the cell to expand to whichever step you like. For the sum, input in B1 (or B2 with a header) : "=SUM(\$A\$1:A1)". Drag down the cell to expand for all values (the $ symbols prevent the cell rows/columns to be moved when being dragged down).}

^{2 : More accurately 0.018696 feet. But physics experts will scream at me that I'm using too much decimal places :p.}

Answer 2

Let's keep things rational.

We're looking at a line from (0,150) (the top of the pyramid) to (200,0) (the edge of the pyramid). This supposes the walkway is in the middle of a side.

Assuming the steps poke out from the pyramid, the first goes straight up from (200,0) to (200,3), then straight across to (196,3) (going 4/3 as far horizontally to keep the 200/150 ratio). We multiply each of the next changes by .95 to get the successive steps.

Now there are two questions to answer. First, how tall is the 100th step? We take 0.95^99 -- and multiply by 3 feet! -- to get our answer, 18.7 millifeet = 0.224 inches. Next is the unasked question: can we get there from here? For an infinite number of steps we find the sum of 3+3r+3r^2 ... = 3/(1-r). Since r=0.95 that comes out to 60 feet, which means you could climb an infinite number of these steps and never get above 60 feet up (and 80 feet in). So the answer above is reachable.

Answer 3

Your pyramid is 60 feet tall

The first step is $3$ feet.

The second step is $3\cdot 0.95 = 2.85$ feet.

The third step is $3\cdot (0.95)^2 = 2.7075$ feet

$\vdots$

The $n$-th step is $3\cdot (0.95)^n $ feet

The total height is $$3 \cdot (1+0.95 +0.95^2 + \ldots +0.95^{100}) $$ $$= 3 \sum_{n=1}^{100}0.95^n$$

which adds up to about 60 feet.

The picture shows $100$ steps is almost as tall as infinitely many steps. So we cannot solve the problem by adding more steps.

How to fix the pyramid

1: Just enlarge the whole thing by a factor or $150/60 =2.5$. That means the first step is now $7.5$ feet tall and the second is $7.5 \cdot 0.95$ and so forth.

2: Make the steps shrink slower. For example $0.9839$ seems to do the trick.

Click here for interactive graph.

Answer 4

No

Your math is incorrect. The slope of the side of your pyramid is not constant, so the equations you're using are not appropriate.