# Two corners of a triangle have angles of #(3 pi ) / 4 # and # pi / 12 #. If one side of the triangle has a length of #5 #, what is the longest possible perimeter of the triangle?

Largest possible perimeter 28.3196

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To find the longest possible perimeter of the triangle, we need to consider the sum of the lengths of the other two sides.

Given that one side of the triangle has a length of 5, let's denote the other two sides as a and b.

Using the law of cosines, we can find the lengths of these sides.

The law of cosines states:

c^2 = a^2 + b^2 - 2ab * cos(C)

Where c is the side opposite the angle C.

Let's denote the angles given as A = (3π)/4 and B = π/12.

We need to find the side lengths a and b.

For angle A: c^2 = a^2 + 5^2 - 2 * a * 5 * cos((3π)/4)

For angle B: c^2 = b^2 + 5^2 - 2 * b * 5 * cos(π/12)

We need to maximize the perimeter, which is given by P = 5 + a + b.

To find the maximum perimeter, we need to find the maximum values of a and b. This occurs when cos((3π)/4) = -sqrt(2)/2 and cos(π/12) = sqrt(3)/2.

Solving the equations for a and b:

For angle A: c^2 = a^2 + 25 + 10a/sqrt(2)

For angle B: c^2 = b^2 + 25 + 10b*sqrt(3)/2

Now, maximize a and b. We do this by setting the derivatives of the equations with respect to a and b to zero.

After solving these equations, we find the maximum values for a and b.

Then, we calculate the perimeter P = 5 + a + b using these maximum values of a and b.

This will give us the longest possible perimeter of the triangle.

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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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