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°
65
°
? Round to the nearest hundredth.(1 point)
Responses

428.90 ft
428.90 ft

93.26 ft
93.26 ft

178.57 ft
178.57 ft

23.66 ft

Answer 1

To find the distance from the observer to the base of the tower, we can use trigonometry, specifically the tangent function, which relates the angle of elevation to the opposite and adjacent sides of a right triangle.

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

Using the tangent function:

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

We can rearrange this to solve for the adjacent side:

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

Substituting in the values:

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

Now, we calculate \( \tan(65°) \):

\[
\tan(65°) \approx 2.1445
\]

Now, we can plug this value into our equation:

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

Rounding to the nearest hundredth, the distance from the observer to the base of the tower is approximately **93.26 ft**.

So the correct response is:

**93.26 ft**