# Two corners of a triangle have angles of # (5 pi )/ 8 # and # ( pi ) / 4 #. If one side of the triangle has a length of # 14 #, what is the longest possible perimeter of the triangle?

To get the largest possible are, smallest angle should correspond to the side of length 14

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To find the longest possible perimeter of the triangle, we need to determine the length of the third side and then calculate the perimeter using all three sides.

Using the Law of Sines, we can find the length of the third side (denoted by ( c )) using the given angles and side:

[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} ]

Given: ( A = \frac{5\pi}{8} ), ( B = \frac{\pi}{4} ), ( a = 14 ).

We can find ( C ) using the fact that the sum of angles in a triangle is ( \pi ) radians:

[ C = \pi - A - B ]

Then, we can find the length of side ( c ):

[ c = \frac{a \cdot \sin(C)}{\sin(A)} ]

Once we have the length of all three sides, we can calculate the perimeter by adding them together.

After calculating, we find the longest possible perimeter to be approximately ( 36.86 ).

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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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