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.

Answer 1

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

In this case:
- The height of the tower is the opposite side (\( h = 200 \) feet).
- The distance from the observer to the base of the tower is the adjacent side (\( d \)).
- The angle of elevation is \( \theta = 65^\circ \).

The tangent function is defined as:

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

Substituting the known values:

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

To find \( d \), we rearrange the equation:

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

Now we need to calculate \( \tan(65^\circ) \). Using a calculator (make sure it's in degree mode):

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

Now we can substitute this back into the equation for \( d \):

\[
d = \frac{200}{2.1445} \approx 93.23
\]

Thus, the distance from the observer to the base of the tower is approximately:

\[
\boxed{93.23} \text{ feet}
\]