Question
A six-feet-tall man looks off the roof of a five-story hotel. He sees a statue that is 75 ft. away from the hotel. If he is looking at the base of the statue, what angle does his sight line form with the side of the hotel? Assume that each complete story of the hotel is 12 ft. tall.
(1 point)
Responses
32.6°
32.6°
43.9°
43.9°
48.7°
48.7°
54.2°
(1 point)
Responses
32.6°
32.6°
43.9°
43.9°
48.7°
48.7°
54.2°
Answers
GPT-4o mini
To find the angle that the man's line of sight forms with the side of the hotel, we first need to determine the height of the man's eye level when he's looking off the roof of the hotel.
1. **Calculate the height of the hotel:**
Each complete story of the hotel is 12 ft tall and there are 5 stories:
\[
\text{Height of hotel} = 5 \, \text{stories} \times 12 \, \text{ft/story} = 60 \, \text{ft}
\]
2. **Determine the total height from which the man is looking:**
The man is 6 ft tall, so his eye level when standing on the roof of the hotel is:
\[
\text{Total height} = \text{Height of hotel} + \text{height of man} = 60 \, \text{ft} + 6 \, \text{ft} = 66 \, \text{ft}
\]
3. **Set up the right triangle:**
- The vertical leg of the triangle is 66 ft (the height from which he is looking).
- The horizontal leg of the triangle is 75 ft (the distance from the hotel to the base of the statue).
4. **Use the tangent function to find the angle:**
The tangent of the angle \( \theta \) formed with the side of the hotel is given by:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{66 \, \text{ft}}{75 \, \text{ft}}
\]
5. **Calculate \( \tan(\theta) \):**
\[
\tan(\theta) = \frac{66}{75} \approx 0.88
\]
6. **Find \( \theta \) using the arctangent function:**
\[
\theta = \tan^{-1}(0.88)
\]
Using a calculator:
\[
\theta \approx 41.4^\circ
\]
Given the options provided, the closest angle appears to be:
**43.9°**
So, the correct response based on the information given is:
**43.9°**
1. **Calculate the height of the hotel:**
Each complete story of the hotel is 12 ft tall and there are 5 stories:
\[
\text{Height of hotel} = 5 \, \text{stories} \times 12 \, \text{ft/story} = 60 \, \text{ft}
\]
2. **Determine the total height from which the man is looking:**
The man is 6 ft tall, so his eye level when standing on the roof of the hotel is:
\[
\text{Total height} = \text{Height of hotel} + \text{height of man} = 60 \, \text{ft} + 6 \, \text{ft} = 66 \, \text{ft}
\]
3. **Set up the right triangle:**
- The vertical leg of the triangle is 66 ft (the height from which he is looking).
- The horizontal leg of the triangle is 75 ft (the distance from the hotel to the base of the statue).
4. **Use the tangent function to find the angle:**
The tangent of the angle \( \theta \) formed with the side of the hotel is given by:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{66 \, \text{ft}}{75 \, \text{ft}}
\]
5. **Calculate \( \tan(\theta) \):**
\[
\tan(\theta) = \frac{66}{75} \approx 0.88
\]
6. **Find \( \theta \) using the arctangent function:**
\[
\theta = \tan^{-1}(0.88)
\]
Using a calculator:
\[
\theta \approx 41.4^\circ
\]
Given the options provided, the closest angle appears to be:
**43.9°**
So, the correct response based on the information given is:
**43.9°**