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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