Answer: the length of x is approximately 128.4 feet (rounded to one decimal place).
Explanation:
To find the length of x, we can use the Law of Cosines, which states:
c^2 = a^2 + b^2 - 2ab * cos(C)
where c is the side opposite angle C in a triangle.
In this case, we have:
a = 84 feet
b = 97 feet
c = 43 feet (this is the side we want to find, x)
Let's substitute these values into the Law of Cosines equation:
x^2 = 84^2 + 97^2 - 2 * 84 * 97 * cos(C)
Now, we need to find the value of the angle C. To do that, we can use the Law of Sines, which states:
sin(C) = (c / b) * sin(B)
where B is the angle opposite side b.
We have:
c = 43 feet
b = 97 feet
sin(C) = (43 / 97) * sin(B)
Now, we need to find the value of sin(B). To do that, we can use the Law of Sines again:
sin(B) = (b / a) * sin(A)
where A is the angle opposite side a.
We have:
b = 97 feet
a = 84 feet
sin(B) = (97 / 84) * sin(A)
Now, we need to find the value of sin(A). To do that, we can use the Law of Sines one more time:
sin(A) = (a / c) * sin(C)
We have:
a = 84 feet
c = 43 feet
sin(A) = (84 / 43) * sin(C)
Now, we can substitute sin(A) and sin(B) into the previous equations to find sin(C):
sin(C) = (43 / 97) * [(97 / 84) * (84 / 43) * sin(C)]
sin(C) = 1
Now that we have sin(C) = 1, we can find the value of cos(C):
cos(C) = √(1 - sin^2(C)) = √(1 - 1) = 0
Now, let's go back to the Law of Cosines equation and solve for x:
x^2 = 84^2 + 97^2 - 2 * 84 * 97 * cos(C)
x^2 = 84^2 + 97^2 - 2 * 84 * 97 * 0
x^2 = 7056 + 9409
x^2 = 16465
x ≈ √16465 ≈ 128.37 feet (rounded to one decimal place)
Therefore, the length of x is approximately 128.4 feet (rounded to one decimal place).