A triangle has two corners with angles of # pi / 4 # and # (5 pi )/ 8 #. If one side of the triangle has a length of #7 #, what is the largest possible area of the triangle?

Answer 1

#A = 41.87#

Given two angles, the third one in a triangle is fixed. In this case it is #pi/8#. The shortest side length will be opposite the smallest angle, which is this one in this case. Therefore, we know that the side of length 7 is opposite the #pi/8# corner.

With three angles and a side now, we can use the Law of Sines to calculate the other sides and, ultimately, the area's height. https://tutor.hix.ai

#a/(sin(2pi/8)) = c/sin C = 7/(sin(pi/8))# #b/(sin(5pi/8)) = c/sin C = 7/(sin(pi/8))#
#a xx (sin(pi/8)) = 7 xx (sin(2pi/8))#
#b xx (sin(pi/8)) = 7 xx (sin(5pi/8))#
#a xx 0.383 = 7 xx 0.707# ; #a = 12.92# #b xx 0.383 = 7 xx 0.924# ; #b = 16.89#
We now use Heron's formula for the area: #A = sqrt(s(s-a)(s-b)(s-c))#
where # s= (a+b+c)/2# or #"perimeter"/2#. https://tutor.hix.ai https://tutor.hix.ai
# s= (7 + 12.92 + 16.89)/2 = 18.41# #A = sqrt(18.41(18.41 - 7)(18.41 - 12.92)(18.41 - 16.89))#
#A = sqrt(18.41(11.41)(5.49)(1.52)# #A = sqrt(1752.9) = 41.87#
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Answer 2

To find the largest possible area of the triangle, we use the formula for the area of a triangle:

[ A = \frac{1}{2} \times \text{base} \times \text{height} ]

We're given one side length of the triangle, which is 7 units. Let's denote this side as ( c ). We need to find the lengths of the other two sides. We can use the Law of Sines to find these side lengths. The Law of Sines states:

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

Given the angles ( \frac{\pi}{4} ) and ( \frac{5\pi}{8} ), we can find the third angle by subtracting the sum of the other two angles from ( \pi ) (since the sum of angles in a triangle is ( \pi )). Then we can use the Law of Sines to find the ratios of the side lengths.

Once we have the ratios of the side lengths, we can find the lengths of the other two sides by multiplying them by ( c ).

Once we have all three side lengths, we can find the area of the triangle using Heron's formula:

[ A = \sqrt{s(s-a)(s-b)(s-c)} ]

where ( s ) is the semi-perimeter of the triangle, given by:

[ s = \frac{a+b+c}{2} ]

Substitute the lengths of the sides into Heron's formula to find the area of the triangle. Repeat this process for various values of ( a ) and ( b ) until you find the maximum area.

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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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