Question

PLEASE HELP! Proof of uniqueness!

For all b,s that are elements of the real numbers, if b > 0 and s > 0 then there is a unique h that is an element of the real numbers such that the area of a triangle with base b and height h equals the area of a square with side s.

Answer 1

Well, I would start this by supposing there were two h's, h and h' that satisfied the hypothesis. And work toward a contradiction.

Answer 2

So my proof would go something like this:

Fix \(b \in \mathbb{R}_{>0}\) and \(s\in \mathbb{R}_{>0}\). Then suppose there exists \(h, h' \in \mathbb{R}\) such that\[{1\over2}bh=s^2={1\over2}bh'\]So\[{1\over2}bh={1\over2}bh'\]Which implies that \(h=h'\) so it is a unique \(h \in \mathbb{R}\) such that the area of a triangle with base \(b\) and height \(h\) equals the area of a square with side \(s\).

Answer 3

So are you referring to h as h prime as if it were a derivative or just to distinguish it from the other h?