Question
Kehlani is trying to find the height of a radio antenna on the roof of a local building. She stands at a horizontal distance of 24 meters from the building. The angle of elevation from her eyes to the roof left parenthesis(point AAright parenthesis) is 20degrees
∘
, and the angle of elevation from her eyes to the top of the antenna left parenthesis(point BBright parenthesis) is 41degrees
∘
. If her eyes are 1.51 meters from the ground, find the height of the antenna left parenthesis(the distance from point AA to point BBright parenthesis). Round your answer to the nearest tenth of a meter if necessary.
∘
, and the angle of elevation from her eyes to the top of the antenna left parenthesis(point BBright parenthesis) is 41degrees
∘
. If her eyes are 1.51 meters from the ground, find the height of the antenna left parenthesis(the distance from point AA to point BBright parenthesis). Round your answer to the nearest tenth of a meter if necessary.
Answers
Answered by
GPT-4o mini
To find the height of the radio antenna, we need to determine the heights of points A (the roof of the building) and B (the top of the antenna) from Kehlani's eye level, and then calculate the height of the antenna by subtracting the height at point A from the height at point B.
1. **Calculating the height of point A:**
The angle of elevation to point A is 20 degrees, and the horizontal distance from Kehlani to the building is 24 meters. We can use the tangent function to find the height of point A above her eye level.
\[
\tan(20^\circ) = \frac{h_A}{24}
\]
Rearranging gives:
\[
h_A = 24 \cdot \tan(20^\circ)
\]
Now we can calculate \(h_A\):
\[
h_A = 24 \cdot \tan(20^\circ) \approx 24 \cdot 0.36397 \approx 8.736 \text{ meters}
\]
Therefore, the height of point A above her eye level is approximately 8.736 meters.
2. **Calculating the height of point B:**
The angle of elevation to point B is 41 degrees. Using the same approach:
\[
\tan(41^\circ) = \frac{h_B}{24}
\]
Rearranging gives:
\[
h_B = 24 \cdot \tan(41^\circ)
\]
Calculating \(h_B\):
\[
h_B = 24 \cdot \tan(41^\circ) \approx 24 \cdot 0.86929 \approx 20.83 \text{ meters}
\]
Therefore, the height of point B above her eye level is approximately 20.83 meters.
3. **Calculating the height of the antenna:**
The height of the antenna (the distance from point A to point B) can be found by subtracting the height of point A from the height of point B:
\[
\text{Height of antenna} = h_B - h_A \approx 20.83 - 8.736 \approx 12.094 \text{ meters}
\]
4. **Final height of the antenna in meters:**
Rounding to the nearest tenth:
\[
\text{Height of antenna} \approx 12.1 \text{ meters}
\]
Thus, the height of the radio antenna is approximately **12.1 meters**.
1. **Calculating the height of point A:**
The angle of elevation to point A is 20 degrees, and the horizontal distance from Kehlani to the building is 24 meters. We can use the tangent function to find the height of point A above her eye level.
\[
\tan(20^\circ) = \frac{h_A}{24}
\]
Rearranging gives:
\[
h_A = 24 \cdot \tan(20^\circ)
\]
Now we can calculate \(h_A\):
\[
h_A = 24 \cdot \tan(20^\circ) \approx 24 \cdot 0.36397 \approx 8.736 \text{ meters}
\]
Therefore, the height of point A above her eye level is approximately 8.736 meters.
2. **Calculating the height of point B:**
The angle of elevation to point B is 41 degrees. Using the same approach:
\[
\tan(41^\circ) = \frac{h_B}{24}
\]
Rearranging gives:
\[
h_B = 24 \cdot \tan(41^\circ)
\]
Calculating \(h_B\):
\[
h_B = 24 \cdot \tan(41^\circ) \approx 24 \cdot 0.86929 \approx 20.83 \text{ meters}
\]
Therefore, the height of point B above her eye level is approximately 20.83 meters.
3. **Calculating the height of the antenna:**
The height of the antenna (the distance from point A to point B) can be found by subtracting the height of point A from the height of point B:
\[
\text{Height of antenna} = h_B - h_A \approx 20.83 - 8.736 \approx 12.094 \text{ meters}
\]
4. **Final height of the antenna in meters:**
Rounding to the nearest tenth:
\[
\text{Height of antenna} \approx 12.1 \text{ meters}
\]
Thus, the height of the radio antenna is approximately **12.1 meters**.