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)Responses23.66 ft.23.66 ft.178.57 ft.178.57 ft.428.90 ft. 428.90 ft. 93.26 ft.

Answer 1

To find the distance from the observer to the base of the tower, we can use the tangent function in trigonometry. The tangent of an angle in a right triangle is the ratio of the opposite side (the height of the tower) to the adjacent side (the distance from the observer to the base of the tower).

Let \( h = 200 \) feet (height of the tower) and let \( d \) be the distance from the observer to the base of the tower. We are given that the angle of elevation is \( \theta = 65^\circ \).

The tangent function gives us:

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

Substituting the values into the equation:

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

Now, we can solve for \( d \):

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

Using a calculator, we find:

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

Now substituting back into the equation for \( d \):

\[
d \approx \frac{200}{2.1445} \approx 93.28 \text{ feet}
\]

Rounding to the nearest hundredth, the distance \( d \) is approximately:

\[
\boxed{93.26 \text{ ft}}
\]