A triangle has two corners with angles of # pi / 12 # and # (5 pi )/ 8 #. If one side of the triangle has a length of #11 #, what is the largest possible area of the triangle?

Answer 1

Largest possible area of the triangle is 171.333

Given are the two angles #(5pi)/8# and #pi/12# and the length 11

The remaining angle:

#= pi - (((5pi)/8) + pi/12) = (7pi)/24#

I am assuming that length AB (11) is opposite the smallest angle.

Using the ASA

Area#=(c^2*sin(A)*sin(B))/(2*sin(C)#
Area#=( 11^2*sin((7pi)/24*sin((5pi))/8))/(2*sin(pi/12))#
Area#=171.333#
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Answer 2

To find the largest possible area of the triangle, we can use the formula for the area of a triangle given two sides and the included angle:

Area = (1/2) * a * b * sin(C)

Where 'a' and 'b' are the lengths of the two sides and C is the included angle.

Given that one side of the triangle has a length of 11 and the angles are π/12 and (5π)/8, we can find the area by using the formula. To maximize the area, we need to use the largest angle possible.

Let's denote the sides of the triangle as a, b, and c. We know that one side, let's say side 'a', has a length of 11. The other two sides are denoted as 'b' and 'c'. The included angle between sides 'b' and 'c' is (5π)/8, which is the largest angle given.

We can use the Law of Sines to find the lengths of sides 'b' and 'c':

sin(A)/a = sin(B)/b = sin(C)/c

Since we know the length of side 'a' and angle 'A', we can find the length of side 'b':

sin(A) = sin(π/12) ≈ 0.2588 sin(C) = sin(5π/8) ≈ 0.9239

So, we have: 0.2588 / 11 = 0.9239 / b

Solving for 'b', we get: b ≈ (11 * 0.9239) / 0.2588 ≈ 39.217

Now, we can calculate the area of the triangle using the formula:

Area = (1/2) * a * b * sin(C) = (1/2) * 11 * 39.217 * sin(5π/8) ≈ (1/2) * 11 * 39.217 * 0.9239 ≈ 204.616

Therefore, the largest possible area of the triangle is approximately 204.616 square units.

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Answer from HIX Tutor

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.

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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