Question
Isaiah is trying to find the height of a radio antenna on the roof of a local building. He stands at a horizontal distance of 20 meters from the building. The angle of elevation from his eyes to the roof left parenthesis(point AAright parenthesis) is 32degrees
∘
, and the angle of elevation from his eyes to the top of the antenna left parenthesis(point BBright parenthesis) is 43degrees
∘
. If his eyes are 1.62 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 meter if necessary.
Answer
Attempt 1 out of 2
∘
, and the angle of elevation from his eyes to the top of the antenna left parenthesis(point BBright parenthesis) is 43degrees
∘
. If his eyes are 1.62 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 meter if necessary.
Answer
Attempt 1 out of 2
Answers
Answered by
GPT-4o mini
To find the height of the antenna, we need to calculate the heights of both points A (the roof of the building) and B (the top of the antenna) based on the angle of elevation from Isaiah's eyes.
Let’s denote:
- \( h_A \) = height of the roof (point A) above Isaiah’s eye level,
- \( h_B \) = height of the top of the antenna (point B) above Isaiah’s eye level,
- \( d = 20 \) meters (the horizontal distance from Isaiah to the building),
- \( \theta_A = 32^\circ \) (angle of elevation to the roof),
- \( \theta_B = 43^\circ \) (angle of elevation to the top of the antenna).
### Step 1: Calculate height \( h_A \)
Using the tangent function:
\[
h_A = d \cdot \tan(\theta_A)
\]
Substituting the values:
\[
h_A = 20 \cdot \tan(32^\circ)
\]
Calculating \( \tan(32^\circ) \):
\[
\tan(32^\circ) \approx 0.6249
\]
So,
\[
h_A = 20 \cdot 0.6249 \approx 12.498 \text{ meters}
\]
### Step 2: Calculate height \( h_B \)
Using the tangent function again:
\[
h_B = d \cdot \tan(\theta_B)
\]
Substituting the values:
\[
h_B = 20 \cdot \tan(43^\circ)
\]
Calculating \( \tan(43^\circ) \):
\[
\tan(43^\circ) \approx 0.9325
\]
So,
\[
h_B = 20 \cdot 0.9325 \approx 18.65 \text{ meters}
\]
### Step 3: Calculate the height of the antenna
The height of the antenna \( H \) is the difference between the two heights:
\[
H = h_B - h_A
\]
Substituting the values:
\[
H = 18.65 - 12.498 \approx 6.152 \text{ meters}
\]
### Step 4: Calculate total height above the ground
Now, since we are required to find the height from the ground, we need to add Isaiah’s eye level height (1.62 meters) to both heights:
The height of the building above ground is:
\[
h_A + 1.62 = 12.498 + 1.62 \approx 14.118 \text{ meters}
\]
The height of the antenna above the ground is:
\[
h_B + 1.62 = 18.65 + 1.62 \approx 20.27 \text{ meters}
\]
### Final Calculation
To find the height of the antenna above the roof:
\[
\text{Height of the antenna} = H \approx 6.152 \text{ meters}
\]
### Conclusion
Rounding this to the nearest meter gives:
\[
\text{Height of the antenna} \approx 6 \text{ meters}
\]
The height of the antenna above the roof of the building, rounded to the nearest meter, is **6 meters**.
Let’s denote:
- \( h_A \) = height of the roof (point A) above Isaiah’s eye level,
- \( h_B \) = height of the top of the antenna (point B) above Isaiah’s eye level,
- \( d = 20 \) meters (the horizontal distance from Isaiah to the building),
- \( \theta_A = 32^\circ \) (angle of elevation to the roof),
- \( \theta_B = 43^\circ \) (angle of elevation to the top of the antenna).
### Step 1: Calculate height \( h_A \)
Using the tangent function:
\[
h_A = d \cdot \tan(\theta_A)
\]
Substituting the values:
\[
h_A = 20 \cdot \tan(32^\circ)
\]
Calculating \( \tan(32^\circ) \):
\[
\tan(32^\circ) \approx 0.6249
\]
So,
\[
h_A = 20 \cdot 0.6249 \approx 12.498 \text{ meters}
\]
### Step 2: Calculate height \( h_B \)
Using the tangent function again:
\[
h_B = d \cdot \tan(\theta_B)
\]
Substituting the values:
\[
h_B = 20 \cdot \tan(43^\circ)
\]
Calculating \( \tan(43^\circ) \):
\[
\tan(43^\circ) \approx 0.9325
\]
So,
\[
h_B = 20 \cdot 0.9325 \approx 18.65 \text{ meters}
\]
### Step 3: Calculate the height of the antenna
The height of the antenna \( H \) is the difference between the two heights:
\[
H = h_B - h_A
\]
Substituting the values:
\[
H = 18.65 - 12.498 \approx 6.152 \text{ meters}
\]
### Step 4: Calculate total height above the ground
Now, since we are required to find the height from the ground, we need to add Isaiah’s eye level height (1.62 meters) to both heights:
The height of the building above ground is:
\[
h_A + 1.62 = 12.498 + 1.62 \approx 14.118 \text{ meters}
\]
The height of the antenna above the ground is:
\[
h_B + 1.62 = 18.65 + 1.62 \approx 20.27 \text{ meters}
\]
### Final Calculation
To find the height of the antenna above the roof:
\[
\text{Height of the antenna} = H \approx 6.152 \text{ meters}
\]
### Conclusion
Rounding this to the nearest meter gives:
\[
\text{Height of the antenna} \approx 6 \text{ meters}
\]
The height of the antenna above the roof of the building, rounded to the nearest meter, is **6 meters**.