Chapter 8.3, Problem 50AYU

Finding the Length of a Guy Wire A radio tower 500 feet high is located on the side of a hill with an inclination to the horizontal of $5^{\circ }$. See the figure. How long should two guy wires be if they are to connect to the top of the tower and be secured at two points 100 feet directly above and directly below the base of the tower?

To determine

To find: How long should two guy wires be if they are to connect to the top of the tower and be secured at two points 100 feet directly above and directly below the base of the tower?

Answer to Problem 50AYU

The guy wire needs to be about $501.28$ feet and $518.38$ feet long.

Explanation of Solution

Given:

A radio tower 500 feet high is located on the side of a hill with an inclination to the horizontal of $5°$. See the figure.

Formula used:

$c^2=a^2+b^2-2ab\text{cos}C$

Calculation:

Let $x+y$ be length of the guy wire that is secured at two points 100 feet directly above and directly below the base of the tower.

$x²=500²+100²-2*500*100*\text{ cos}\left(85°\right)$

$x=\sqrt{{500}^2+{100}^2-100000\text{cos}85}$

$x\approx 501.28$

$y²=100²+{\left(500\right)}^2-2*100*\left(500\right)\text{cos}95°$

$y=\sqrt{10000+25000-100000\text{cos}95°}$

$y\approx 518.38$

The guy wire needs to be about $501.28$ feet and $518.38$ feet long.