Original Problem! Expanding Galaxies and Rates of Change

Afiq Hatta here presents a nice little problem that mixes geometry and calculus. And it’s inspired by cosmology, to cover an extra subject.

Maths from Afiq

This post was inspired by an article on cosmology that I read, which looked at the possible models of how the universe expanded. So, I created a problem who’s constraints almost mimic that of three expanding galaxies moving away for each other. The problem is as follows:

Expanding Galaxies Problem

Galaxies A and B, B and C, C and A have initial displacements of “a”, “b”, and “c” between them respectively. Given that that the magnitude of the vectors AB, BC, AC are increasing at the same rate, what is the rate of change of the area of the triangle ABC? Express as a function of “phi”, t, and abc.

Tip: you may find the following formula for the area of a triangle useful!

CodeCogsEqn (89)

Where,CodeCogsEqn (90)

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