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 basic trigonometry, specifically the tangent function. The height of the tower and the distance from the observer to the base of the tower form a right triangle, where:

- The height of the tower (opposite side) = 200 feet
- The angle of elevation from the observer to the top of the tower = 65°
- The distance from the observer to the base of the tower (adjacent side) = x

We use the tangent of the angle:

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

Plugging in the values we have:

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

To find x, we can rearrange the equation:

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

Now we need to calculate \(\tan(65^\circ)\):

1. Calculate \(\tan(65^\circ)\) using a calculator:
\[
\tan(65^\circ) \approx 2.1445
\]

2. Now substitute that value back into the equation for x:
\[
x = \frac{200}{2.1445} \approx 93.26
\]

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

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