Help with math functions

Can someone explain to me step by step how to solve this problem please?

I need help solving the following problems:

1. Express the volume of a rectangular solid with height: x width: 2x length: 3x as a function of its width.

2. Using the same rectangular solid above, express the length of a diagonal in terms of its height. (the diagonal passes through the center of the box and connects two corners that do not have a face of the box in common)

thanks for your help in advance.

I know how to find the volume but how do I express it as a function of the width, in this case 2x? I don't just want the solution without an explanation because I want to understand how it is done and be able to do other problems on my own...

Question #1

We need to get a function that we plug in the width of the rectangle (or any rectangle with the same proportions), we'll get the volume.

Let w be the width of our rectangle. You can pick any variable you like (for example, if using "y" makes more sense to you, then do that).
We know: w = 2x
Given this, we also know
height = w/2
length = (3/2)*w

We know the volume is height*width*length so:

V(w) = (w/2)*w*(3/2)*w
Simplifying we get...
V(w) = (3*w^3)/(4)


Question #2

I'm not sure what level of class you're in, so I'll just give you the formula for the diagonal of a rectangle solid: D = (l^2 + w^2 + h^2)^(1/2) = sqrt(l^2 + w^2 + h^2)
l = length
w = width
h = height
D = length of the diagonal

Depending on the class, you may be able to derive said formula on your own. If you're curious, I'm sure you could find a decent website that explains how you do it (you kind of need pictures to do it justice, which is why I'm not going to attempt it here).

Either way, we have the formula and need to express the length of the diagonal in terms of the height. We approach this the same way we approached the first problem, although it's a bit easier because our height is x!