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

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:

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!

Hi Joseph

You need to explain to “The Malaysian Mathematics” that there is a world (universe) of difference between his formulation of the problem, with its constant rates of change for a, b and c, and an expansion, in which the rate of displacement of two points is proportional to the current displacement.

In the constant case the triangle will eventually become equilateral, in the expansion case the shape of the triangle remains the same and its rate of expansion is proportional to the square of the displacement.

You can do the math!

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You are totally correct howardat58 – but it depends on the model of cosmological expansion used! That’s why I said *almost*. I am actually working on a problem which will include a d|AC|/dt = k * AC, and so on. Thank you for your feedback! It is only inspiration, not a following. Sorry once again – I may even edit the point. Of course if you really wanted to go into detail you could include relativity and so forth, depending on what model (Steady state/ etc.) of expansion you wish to cover! Hey if you would like to collaborate….

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I probably ought to have explained more in my introduction that I understood this was a problem inspired by cosmological expansion rather than directly drawn from it. A good problem can come from even an unrealistically loose model.

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Nice post. I like this type of problems because it could make the students more enthusiasts with calculus. I mean, when someone ask to the student: What are you studying in calculus? R/ Something with expanding galaxies.

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I’d like to think so. It’s an imagination-capturing sort of problem, at least, even granting that the relationship to actual galaxy behavior is slight. It’s hard to find problems that are both about stuff people naturally want to know more about and that fit neatly a particular problem to solve.

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