I determined the shape of the crater to be a spherical cap. According to this website, http://mathforum.org/library/drmath/view/55253.html , its formula is described as

$$V_{cap} = \frac{1}{6} \pi h (3c^2 + h^2)$$


$V_{cap}$ = Volume of spherical cap

h = height of the cap

c = radius of the cap

What I want to find out here is what is the relationship between volume of the cap and the radius of the cap. Is $V$ proportional to $h$ or $h^2$?

I can't seem to figure them out.


As per your equation, $$V_{cap} = \frac{1}{6} \pi h(3c^2 + h^2)$$ or in other words, $$V_{cap} = \frac{1}{2} \pi c^2 h + \frac{1}{6} \pi h^3$$

It is observable that $V$ depends on $h$ and also $h^3$, that is, $V$ depends on a cubic relation of $h$.

Now a preferable way to approach this is that:

Consider you are increasing the value of $h$ to very large values. You will notice that $h^3$ starts dominating over $h$, that is, the value of $h$ becomes negligible compared to $h^3$ as $h$ tends to larger and larger values, in crude sense, $h$ tends to infinity.

Looking from this point of view, you can comment that $$V \propto h^3$$

Hope this helps.

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