DG has disagreed with a statement that I made in a previous post where I compared the size of a toad to the size of a dime. I summarize the post below:
You could fit a dozen of those guys on that dime.
It gives new meaning to the phrase “Dime a dozen”.
DG tells me that there is no way I could fit a dozen of those toads on that dime, so the gag is invalid. However, I can prove my estimate using some basic physics. So pay attention, I am only going to do this once.
First, we assume that a toad is a perfect, frictionless sphere.
DG: Wha?
Me: Work with me on this.
First, we assume that a toad is a perfect, frictionless sphere.
Now, we can pile up the toads into several layers on the dime, like they used to pile up cannonballs. Like this:
Using some basic physics equations, we can calculate the total number of toads in the pile from the number of layers in the pile.
We pile up the toads in a pyramid in a square based configuration. If we do that, we could fit at least three layers of toads on the dime. From those assumptions, we can calculate the total number of toads that could be stacked on the dime.
Let Q[n] be the number of toads in the pile if the pile has n layers.
DG: Wha?
Since each layer contains a number of toads that equals the square of the layer number for that layer, the formula for the sum of all the toads in all the layers if you add a layer is
which can also be written as the following cubic equation:
Given the assumption that we can fit at least three layers of toads on the dime, we can substitute the number 3 for n into the cubic equation and we get
Therefore, we can get 14 toads on the dime….. two more than my dozen.
Right?
DG had to think about it.
Finally he said, “Oh yeah? Well if the toads are frictionless, you won’t be able to get even one of ’em to stick to that dime!”
Details details! (sore loser!)