volume question

[mathsbrain]

Encountered a terry lee method for finding volumes of a frustum.
Let h=height from apex to typical slice and x=width of typical slice.
He claims h and x are linearly dependent. and so h=ax+b
But don't we need to prove this? I mean is it that easy to see the linear relationship just because the sides are straight lines?
i have been using similar triangle methods until i saw this linearly dependent method. Would appreciate a clearer explanation regarding this linearly dependent logic.

[RuiAce]

Note that fitting the line and use of similar triangles are BOTH accepted in the HSC.

Also, it should be very intuitively clear that a linear relationship exists between them because the height is obviously vertical, and the side is obviously an ordinary diagonal line. If two things are both lines, they can both be modelled by a linear equation in the x-y plane, so if we merge them we should always be able to find a linear relationship between the two. What more is there to prove?

I made a remark about the linear representation in my notes (the book) and how it works. That I won't discuss here as it's not so easy to explain on the spot.

[mathsbrain]

feel so dumb right now...but still dont see how you can "merge" lines......
Is the remark somewhere on this forum?

[mathsbrain]

actually just saw a slanted pole outside my room and the idea came through......
in any right angled triangle, the shorter sides are linearly dependent to one another and the function that maps them is the hypotenuse.....does that even sound right lol
so in this case, the shorter sides are h and x, and obviously they can turn into a right angled triangle......

[RuiAce]

Merge was probably the incorrect word if we are to interpret it literally.

The idea I was trying to mention (but not explain in depth) was just the fact that we can represent multiple lines on a Cartesian plane together, and observe any linear relationship between two lines at a time.