Question

A helicopter is being used to move a large pipe from a circular cut-out of rock face high up on a cliff. The helicopter drops two ropes that are attached to pipe ends. Given that a=84 feet, b=97 feet, and c=43 feet, what is the length of x? (Round your answer to one decimal place, if necessary.

Answer 1

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).