So, I have a given volume, which must remain constant, and I need to find length, width, and height for a rectangular prism with that volume, scaled to a given ratio (Say, 7:1:1 length:width:height for instance).
Obviously I can take the cube root of the volume as a starting point, for a 1:1:1 ratio rectangular prism (aka a cube of course). But how do I go about taking that value and finding the 7:1:1 dimensions I'm looking for from there?
If you could keep math notation to a minimum in the answer, I'd appreciate it. I'm a dumb programmer with limited formal schooling in math. I like math, but I don't speak it. If you do need to write the answer in math notation, please include an explanation for what it represents in simple arithmetic? Thanks!
Taking the given ratio $7:1:1$ as an example.
Imagine there are $7$ cubes of the same size (to be determined). If one stacks the cubes, they would become a rectangular prism of $7:1:1$ ratio. If the rectangular prism has a volume $V$, then each cube would have a volume $\frac V7$, and the OP knows how to find the dimensions of each cube.
Rewriting the same process in variables. Let $w$ be the width of the rectangular prism, then its length $l=7w$ and its height $h = w$. The given volume $V$ would satisfy $V = l\cdot w\cdot h = 7w^3$, i.e.
$$w = \sqrt[3]{\frac{V}{7}}.$$
(Editing ephemeral and temporary comments into an answer.)