Grace is trying to find the height of a radio antenna on the roof of a local building. She stands at a horizontal distance of 21 meters from the building. The angle of elevation from her eyes to the roof left parenthesis(point AAright parenthesis) is 24degrees
∘
, and the angle of elevation from her eyes to the top of the antenna left parenthesis(point BBright parenthesis) is 42degrees
∘
. If her eyes are 1.54 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.

To find the height of the antenna, we can break the problem into two parts:

1. Calculate the height of the building (point A) from the ground.
2. Calculate the height of the antenna above the building (point B).

We'll use trigonometry to solve for these heights. 

### Step 1: Calculate the height of the building (point A)

Given:
- Distance from Grace to the building, \(d = 21\) meters
- Angle of elevation to the roof (point A), \(\theta_A = 24^\circ\)
- Height of Grace's eyes from the ground, \(h_{\text{eyes}} = 1.54\) meters

Using the tangent function:
\[
\tan(\theta_A) = \frac{h_A}{d}
\]
where \(h_A\) is the height of the building above Grace's eyes. Solving for \(h_A\):
\[
h_A = d \cdot \tan(\theta_A) = 21 \cdot \tan(24^\circ)
\]

Calculating \(h_A\):
\[
\tan(24^\circ) \approx 0.4452
\]
\[
h_A = 21 \cdot 0.4452 \approx 9.3492 \text{ meters}
\]

Adding the height of Grace's eyes to get the total height of the building:
\[
H_A = h_A + h_{\text{eyes}} = 9.3492 + 1.54 = 10.8892 \text{ meters}
\]

### Step 2: Calculate the height of the antenna above the building (point B)

Given:
- Angle of elevation to the top of the antenna (point B), \(\theta_B = 42^\circ\)

Using the tangent function again for the total height from Grace's eyes to the top of the antenna:
\[
\tan(\theta_B) = \frac{H_B - h_{\text{eyes}}}{d}
\]
where \(H_B\) is the total height of the building plus the antenna. Solving for \(H_B - h_{\text{eyes}}\):
\[
H_B - h_{\text{eyes}} = d \cdot \tan(\theta_B) = 21 \cdot \tan(42^\circ)
\]

Calculating \(H_B - h_{\text{eyes}}\):
\[
\tan(42^\circ) \approx 0.9004
\]
\[
H_B - h_{\text{eyes}} = 21 \cdot 0.9004 \approx 18.9084 \text{ meters}
\]

Adding the height of Grace's eyes to get the total height to the top of the antenna:
\[
H_B = 18.9084 + 1.54 = 20.4484 \text{ meters}
\]

### Step 3: Calculate the height of the antenna

The height of the antenna is the difference between the total height to the top of the antenna and the height of the building:
\[
h_{\text{antenna}} = H_B - H_A = 20.4484 - 10.8892 \approx 9.6 \text{ meters}
\]

Thus, the height of the antenna is approximately \(9.6\) meters.

Would you like more details on any part of the solution? Here are some related questions:

1. How is the tangent function used in solving right triangle problems?
2. Why do we add the height of Grace's eyes to our calculations?
3. What other trigonometric functions can be used to solve similar height problems?
4. How can errors in angle measurement affect the final result?
5. Can we use the sine or cosine functions for this problem?
6. How would the solution change if Grace stood further away from the building?
7. What is the importance of rounding in real-world measurements?
8. How would you solve this problem if the angles given were in radians?

**Tip:** Always make sure your calculator is set to the correct unit (degrees or radians) when performing trigonometric calculations.

Learn how to calculate the height of a radio antenna on a building using trigonometry. This problem involves finding heights using angles of elevation and distance, applicable formulas like the tangent function, and practical calculations. Suitable for high school students.

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