Question

A radio tower is 200 feet tall. How far is an observer from the base of the tower if the angle of their line of sight from the ground to the top of the tower is 65 degree? Round to the nearest hundredth.

a. 23.66 ft
b. 178.57 ft
c. 428.90 ft
d. 93.26 ft

Answer 1

To find the distance from the observer to the base of the tower, we can use the tangent function from trigonometry, which relates the angle of elevation to the height of the tower and the distance from the tower.

The relationship is given by:

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]

Where:
- \(\theta\) is the angle of elevation (65 degrees),
- The opposite side is the height of the tower (200 ft),
- The adjacent side is the distance from the observer to the base of the tower (let's call it \(d\)).

Rearranging the formula to solve for \(d\):

\[
d = \frac{\text{opposite}}{\tan(\theta)}
\]

Now, substituting in the values:

\[
d = \frac{200}{\tan(65^\circ)}
\]

Calculating \(\tan(65^\circ)\):

\[
\tan(65^\circ) \approx 2.1445
\]

Now substituting that into the equation:

\[
d = \frac{200}{2.1445} \approx 93.26 \text{ ft}
\]

So, the distance from the observer to the base of the tower is approximately **93.26 ft**.

The correct answer is **d. 93.26 ft**.