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...
Answers
GPT-4o mini
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 and adjacent sides of a right triangle.
Let:
- \( h \) = height of the tower = 200 feet
- \( D \) = distance from the observer to the base of the tower
- \( \theta \) = angle of elevation = 65°
The tangent of the angle is given by the formula:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{D}
\]
We can rearrange this to solve for \( D \):
\[
D = \frac{h}{\tan(\theta)}
\]
Now substituting the values:
\[
D = \frac{200}{\tan(65^\circ)}
\]
Now we'll calculate the tangent of 65 degrees:
\[
\tan(65^\circ) \approx 2.1445
\]
Substituting this value back into the equation for \( D \):
\[
D = \frac{200}{2.1445} \approx 93.24 \text{ feet}
\]
Thus, the distance from the observer to the base of the tower is approximately **93.24 feet**.
Let:
- \( h \) = height of the tower = 200 feet
- \( D \) = distance from the observer to the base of the tower
- \( \theta \) = angle of elevation = 65°
The tangent of the angle is given by the formula:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{D}
\]
We can rearrange this to solve for \( D \):
\[
D = \frac{h}{\tan(\theta)}
\]
Now substituting the values:
\[
D = \frac{200}{\tan(65^\circ)}
\]
Now we'll calculate the tangent of 65 degrees:
\[
\tan(65^\circ) \approx 2.1445
\]
Substituting this value back into the equation for \( D \):
\[
D = \frac{200}{2.1445} \approx 93.24 \text{ feet}
\]
Thus, the distance from the observer to the base of the tower is approximately **93.24 feet**.