
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 (point A) is 32°, and the angle of elevation from his eyes to the top of the antenna (point B ) is 43°. If his eyes are 1.62 meters from the ground, find the height of the antenna (the distance from point A to point B). Round your answer to the nearest meter if necessary.

To solve the problem of finding the height of the radio antenna, we can break it down into two parts:

1. **Calculate the height of the building (up to point A).**
2. **Calculate the total height up to the top of the antenna (point B).**
3. **Subtract the height of the building from the total height to get the height of the antenna.**

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

Let:
- \( h_1 \) be the height of the building above Isaiah's eyes.
- \( d = 20 \) meters be the horizontal distance from Isaiah to the building.
- \( \theta_1 = 32^\circ \) be the angle of elevation from Isaiah's eyes to the roof (point A).

Using the tangent function:
\[
\tan(\theta_1) = \frac{h_1}{d}
\]
\[
h_1 = d \times \tan(32^\circ)
\]

Now, calculate \( h_1 \):
\[
h_1 = 20 \times \tan(32^\circ) \approx 20 \times 0.6249 \approx 12.498 \text{ meters}
\]

The height of the building from the ground (including Isaiah's eye level) is:
\[
H_1 = h_1 + 1.62 \text{ meters} \approx 12.498 + 1.62 \approx 14.118 \text{ meters}
\]

### Step 2: Calculate the total height up to the top of the antenna (point B)

Let:
- \( h_2 \) be the total height above Isaiah's eyes (from his eyes to point B).
- \( \theta_2 = 43^\circ \) be the angle of elevation from Isaiah's eyes to the top of the antenna.

Using the tangent function:
\[
\tan(\theta_2) = \frac{h_2}{d}
\]
\[
h_2 = d \times \tan(43^\circ)
\]

Now, calculate \( h_2 \):
\[
h_2 = 20 \times \tan(43^\circ) \approx 20 \times 0.9325 \approx 18.650 \text{ meters}
\]

The total height from the ground to the top of the antenna is:
\[
H_2 = h_2 + 1.62 \text{ meters} \approx 18.650 + 1.62 \approx 20.270 \text{ meters}
\]

### Step 3: Find the height of the antenna (distance from point A to point B)

Finally, the height of the antenna is:
\[
\text{Height of antenna} = H_2 - H_1 \approx 20.270 - 14.118 \approx 6.152 \text{ meters}
\]

### Final Answer
The height of the antenna is approximately **6 meters**.

Would you like more details or have any questions?

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**Related Questions:**
1. How would the calculation change if the horizontal distance were increased?
2. Can you calculate the total height if Isaiah stood closer to the building?
3. What if the angles were measured from a different eye level?
4. How would the height of the antenna change if the building was taller?
5. What is the effect of a change in angle of elevation on the calculated height?

**Tip:** In trigonometry problems, make sure your calculator is set to the correct mode (degrees or radians) depending on the angle unit given in the problem.

Learn how to calculate the height of a radio antenna using trigonometric principles. This problem involves finding the height of a building and the total height up to the top of the antenna using given angles of elevation. Suitable for high school students.

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