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.