# Two corners of a triangle have angles of #pi / 3 # and # pi / 6 #. If one side of the triangle has a length of #4 #, what is the longest possible perimeter of the triangle?

The maximum perimeter is

And the maximum perimeter is:

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To find the longest possible perimeter of the triangle, we need to maximize the length of the remaining side, which is opposite the angle of π/3.

Using the Law of Sines, we can find the lengths of the other sides:

Let x be the length of the side opposite the angle of π/3.

Using the Law of Sines: sin(π/6) / 4 = sin(π/3) / x

Solving for x: x = (4 * sin(π/3)) / sin(π/6) x = (4 * (√3/2)) / (1/2) x = 4 * √3

Now, we can find the perimeter: Perimeter = 4 + 4 + 4√3 Perimeter = 8 + 4√3

So, the longest possible perimeter of the triangle is 8 + 4√3.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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