Chapter 8.3, Problem 49AYU

Finding the Length of a Guy Wire The height of a radi tower is 500 feet, and the ground on one side of the tower slopes upward at an angle of ${10}^{\circ }$ (see the figure).

a. How long should a guy wire be if it is to connect to the top of the tower and be secured at a point on the slope side 100 feet from the base of the tower?

b. How long should a second guy wire be if it is to connect to the middle of the tower and be secured at a positio 100 feet from the baseon the flat side?

To determine

To find:

a. How long should a guy wire be if it is to connect to the top of the tower and be secured at a point on the sloped side 100 feet from the base of the tower?

Answer to Problem 49AYU

a. Therefore of the guy wire that is secured on the top is $492.6$ ft.

Explanation of Solution

Given:

The height of a radio tower is 500 feet, and the ground on one side of the tower slopes upward at an angle of $10°$ (see the figure).

Formula used:

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

Calculation:

a. Let $x$ be length of the guy wire that is secured on the slope.

$x²=500²+100²-2*500*100*\text{ cos}\left(90°-10°\right)$

$x=\sqrt{{500}^2+{100}^2-100000\text{cos}80}$

$x\approx 492.6$

Therefore of the guy wire that is secured on the top is $492.6$ ft.

To determine

To find:

b. How long should a second guy wire be if it is to connect to the middle of the tower and be secured at a position 100 feet from the base on the flat side?

Answer to Problem 49AYU

b. The length of the guy wire that is secured in the middle is $269.3$ feet.

Explanation of Solution

Given:

The height of a radio tower is 500 feet, and the ground on one side of the tower slopes upward at an angle of $10°$ (see the figure).

Formula used:

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

Calculation:

b. Let $y$ be length of the guy wire that is secured in the middle.

$y²=100²+{\left(\frac{500}{2}\right)}^2-2*100*\left(\frac{500}{2}\right)\cos 90$

$y=\sqrt{10000+62500}$

$y=269.3$

The length of the guy wire that is secured in the middle is $269.3$ feet.