Hello, MmmmSteak!

A 125-ft tower is located on the side of a mountain that is inclined 32° to the horizontal.

A guy wire is to be attached to the top of the tower and anchored at a point 55 ft

downhill from the base of the tower.

Find the shortest length of the wire needed. Code:

o A
* |
* |
* |125
* |
x * 122° o B
* *58°|
* * |
* * 55 |
* * 32° |
C o - - - - - - - - - o D

The tower is: .$\displaystyle AB = 125$

The guy wire is: .$\displaystyle x = AC$

And: .$\displaystyle BC = 55$

$\displaystyle \angle BCD = 32^o$

. . Then: .$\displaystyle \angle CBD = 58^o$

. . Hence: .$\displaystyle \angle ABC = 122^o$

Law of Cosines: .$\displaystyle x^2 \;=\;125^2 + 55^2 - 2(125)(55)\cos122^o \;=\;25936.38988$

Therefore: .$\displaystyle x \;=\;161.0477876 \;\approx\;\boxed{161\text{ ft}} $