SOLUTION: A 119-foot tower is located on a hill that is inclined 32° to the horizontal, as shown in the figure below. A guy wire is to be attached to the top of the tower and anchored at a

Algebra ->  Pythagorean-theorem -> SOLUTION: A 119-foot tower is located on a hill that is inclined 32° to the horizontal, as shown in the figure below. A guy wire is to be attached to the top of the tower and anchored at a       Log On


   

Question 1200405: A 119-foot tower is located on a hill that is inclined 32° to the horizontal, as shown in the figure below. A guy wire is to be attached to the top of the tower and anchored at a point 97 feet uphill from the base of the tower. Find the length of wire needed. (Round your answer to one decimal place.)
Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!

Answer: 106.5 feet
This result is approximate.


Explanation:

x = length of wire needed
This is some positive real number.

The image attachment you posted/sent isn't showing up.
Fortunately the verbal description appears to be sufficient.

From the written description you provided, I'm assuming the diagram looks like this:

Please let me know if my assumption is incorrect.

For the sake of simplicity, we have the following assumptions:
  • The tower is completely vertical.
  • The hill is a straight line.
  • The slope or gradient does not change anywhere along the hill.
Let's draw a red line from the base of the tower and move toward the right a bit.


We can slide the 32 degree angle label up and to the right.
This is because translation/shifting operations preserve the angle measure.
For more information, search out "corresponding angles theorem".

This is what we have when we add that 32 degree label.

The angle adjacent to this new 32 degree angle is 90-32 = 58 degrees. The two angles 32 and 58 are complementary. They form a 90 degree corner.

Let's add that angle label.


There's a bit of clutter, so let's focus on the triangle.
Erase the lines we won't need from this point onward.

At this point, we'll need the Law of Cosines to determine x.

c^2 = a^2 + b^2 - 2*a*b*cos(C)
x^2 = 119^2 + 97^2 - 2*119*97*cos(58)
x^2 = 11336.2838659122
x = sqrt(11336.2838659122)
x = 106.471986296453
x = 106.5 feet of wire is needed (approximate).
Make sure your calculator is in degree mode.