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°
? Round to the nearest hundredth.(1 point)
Responses

23.66 ft.
23.66 ft.

428.90 ft.
428.90 ft.

93.26 ft.
93.26 ft.

178.57 ft.

Answer 1

To determine how far the observer is from the base of the tower, we can use the tangent function from trigonometry. The formula involving the tangent of an angle in a right triangle is:

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

In this scenario:
- The angle \( \theta \) is 65°.
- The height of the tower (opposite side) is 200 feet.
- The distance from the observer to the base of the tower (adjacent side) is what we need to find.

Rearranging the formula gives us:

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

Substituting in the values we have:

\[
\text{adjacent} = \frac{200}{\tan(65°)}
\]

Now, we can calculate \( \tan(65°) \):

Using a calculator, \( \tan(65°) \approx 2.1445 \).

Now, substituting that back into our equation:

\[
\text{adjacent} \approx \frac{200}{2.1445} \approx 93.26 \text{ ft.}
\]

Therefore, the observer is approximately **93.26 feet** from the base of the tower.

So the correct answer is:

**93.26 ft.**