Solution

Notice that in the beginning, the points form an equilateral triangle. By symmetry, they will always form an equilateral triangle. Suppose that at time t the points are in an equilateral triangle with sides a(t). At the time point A will have moved v toward point B, point B will have moved v toward point C, and point C will have moved v toward point A. Call these new points A', B', and C'.

As you can see in the diagram the new side a() can be found by the Law of Cosines.

Then, if we subtract a^2(t) from both sides, factor the left-hand side, and divide both sides by we have

Now if , the first factor on the left-hand side becomes a(t)' by definition of a derivative, the second factor becomes 2a(t) and on the right-hand side, the last addend becomes zero.

a(t) cancels out on both sides, so we integrate to obtain

We know that when t=0, the side a(t) is 1, therefore the constant C = 1. Also it is given that v = 50. We have to find the time t when a = 0.

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