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 solve for the height of the antenna, we need to follow a few steps involving trigonometry. We will first find the height of the building and then the combined height of the building and the antenna. Finally, we subtract the height of the building from the combined height to find the height of the antenna.

1. **Determine the height of the building (h1):**

   Given:
   - Distance from Grace to the building (d) = 21 meters
   - Angle of elevation to the roof of the building (A) = 24°
   - Height of Grace's eyes from the ground (h_eye) = 1.54 meters

   The total height from Grace's eyes to the roof of the building can be calculated using the tangent function:
   \[
   \tan(24^\circ) = \frac{\text{Height from eyes to roof (h_1)}}{\text{Distance to the building (d)}}
   \]
   \[
   \tan(24^\circ) = \frac{h_1}{21}
   \]
   Solving for \(h_1\):
   \[
   h_1 = 21 \times \tan(24^\circ)
   \]

   Using a calculator:
   \[
   \tan(24^\circ) \approx 0.4452
   \]
   \[
   h_1 = 21 \times 0.4452 \approx 9.3492 \text{ meters}
   \]

   The height of the building from the ground is then:
   \[
   h_{building} = h_1 + h_{eye} = 9.3492 + 1.54 = 10.8892 \text{ meters}
   \]

2. **Determine the height from Grace's eyes to the top of the antenna (h2):**

   Given:
   - Angle of elevation to the top of the antenna (B) = 42°

   Similarly, we use the tangent function:
   \[
   \tan(42^\circ) = \frac{\text{Height from eyes to top of antenna (h_2)}}{\text{Distance to the building (d)}}
   \]
   \[
   \tan(42^\circ) = \frac{h_2}{21}
   \]
   Solving for \(h_2\):
   \[
   h_2 = 21 \times \tan(42^\circ)
   \]

   Using a calculator:
   \[
   \tan(42^\circ) \approx 0.9004
   \]
   \[
   h_2 = 21 \times 0.9004 \approx 18.9084 \text{ meters}
   \]

   The height of the antenna from the ground is:
   \[
   h_{total} = h_2 + h_{eye} = 18.9084 + 1.54 = 20.4484 \text{ meters}
   \]

3. **Determine the height of the antenna (h_antenna):**

   Subtract the height of the building from the height to the top of the antenna:
   \[
   h_{antenna} = h_{total} - h_{building} = 20.4484 - 10.8892 = 9.5592 \text{ meters}
   \]

Rounding to the nearest tenth:
\[
h_{antenna} \approx 9.6 \text{ meters}
\]

Thus, the height of the antenna is approximately **9.6 meters**.

Would you like more details or have any questions?

Here are 8 related questions:
1. How do you calculate the height of an object using trigonometry?
2. What is the tangent function and how is it used in these calculations?
3. Can you explain the significance of the angle of elevation in these problems?
4. How does the height of the observer's eyes affect the overall height calculation?
5. What are some other real-world applications of trigonometry?
6. How would the problem change if the observer's height was different?
7. What are the other trigonometric functions, and how are they used?
8. How can you verify the accuracy of your trigonometric calculations?

**Tip:** Always ensure your calculator is set to the correct mode (degrees or radians) depending on the units of the angles you are working with.

Learn how to calculate the height of an antenna using trigonometry with angles of elevation 24° and 42°. Follow a step-by-step solution to find the height of the antenna from given distances and angles. Suitable for high school students in Grades 10-12.

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