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)
Responses
428.90 ft.
428.90 ft.
178.57 ft.
178.57 ft.
93.26 ft.
93.26 ft.
23.66 ft.
? Round to the nearest hundredth.(1 point)
Responses
428.90 ft.
428.90 ft.
178.57 ft.
178.57 ft.
93.26 ft.
93.26 ft.
23.66 ft.
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 side (the height of the tower) and the adjacent side (the distance from the observer to the base of the tower).
We can set up the equation using the tangent of the angle:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
In this case, \(\theta = 65^\circ\), the opposite side (height of the tower) is 200 feet, and the adjacent side (the distance we want to find) is \(d\):
\[
\tan(65^\circ) = \frac{200}{d}
\]
Rearranging the equation to solve for \(d\):
\[
d = \frac{200}{\tan(65^\circ)}
\]
Calculating \(\tan(65^\circ)\) using a calculator:
\[
\tan(65^\circ) \approx 2.1445
\]
Now substituting this value into the equation:
\[
d \approx \frac{200}{2.1445} \approx 93.26 \text{ ft}
\]
Therefore, the distance from the observer to the base of the tower is approximately \(93.26\) feet.
The correct response is:
**93.26 ft.**
We can set up the equation using the tangent of the angle:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
In this case, \(\theta = 65^\circ\), the opposite side (height of the tower) is 200 feet, and the adjacent side (the distance we want to find) is \(d\):
\[
\tan(65^\circ) = \frac{200}{d}
\]
Rearranging the equation to solve for \(d\):
\[
d = \frac{200}{\tan(65^\circ)}
\]
Calculating \(\tan(65^\circ)\) using a calculator:
\[
\tan(65^\circ) \approx 2.1445
\]
Now substituting this value into the equation:
\[
d \approx \frac{200}{2.1445} \approx 93.26 \text{ ft}
\]
Therefore, the distance from the observer to the base of the tower is approximately \(93.26\) feet.
The correct response is:
**93.26 ft.**