Chapter 8.3, Problem 49AYU

a.

To determine

Find the length of guy wire?

Answer to Problem 49AYU

  $\boxed{\text{492}.\text{6}}$

Explanation of Solution

Given information:

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

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?

Calculation:

From the figure, we can say that the angle between the tower and the gound is

  $90°-10°=80°$

Let the length of the guy wire be $x$ .

Since we know the two sides and the included angle of the triangle, the Law of Cosines can be applied to find the length of guy wire.

  $x^2={500}^2+{100}^2-\mathrm{2.500.100.}\cos 80°$

  $\begin{array}{l}\approx 250000+10000-17365\\ =242635\end{array}$

Take square root on both the sides.

  $\begin{array}{l}\sqrt{x^2}=\sqrt{242635}\\ x\approx 492.6\end{array}$

Hence, the guy wire should be about 492.6 ft long.

b.

To determine

Find the length of second guy wire?

Answer to Problem 49AYU

  $\boxed{269.3ft}$

Explanation of Solution

Given information:

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

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

Calculation:

Let the length of the guy wire connected to the middle of the tower be y.

Since the wire is connected to the middle of the tower, one side of the triangle formed will be $\frac{500}{2}=250ft$ .

Use the Law of Cosines to find $y$ .

  $y^2={250}^2+{100}^2-\mathrm{2.250.100.}\cos 90°$

  $=62500+10000-0$

  $=72500$

Take square root on both the sides.

  $\sqrt{y^2}=\sqrt{72500}$

  $y\approx 269.3$

Hence, the guy wire should be about $\boxed{269.3ft}$ long.