To find the height of the antenna, we need to determine the height of the building first and then the total height up to the top of the antenna. Finally, we subtract the building height from the total height to get the antenna height.

Step 1: Define Given Information

• Distance from Isaiah to the base of the building: 20 meters
• Angle of elevation to the roof of the building (Point A): 32°
• Angle of elevation to the top of the antenna (Point B): 43°
• Eye level height: 1.62 meters

Let:

• $h_A$ be the height of the building above Isaiah's eyes.
• $h_B$ be the height of the antenna above Isaiah's eyes.
• $H_A$ be the total height of the building (including the height from the ground).
• $H_B$ be the total height of the building and antenna.

Step 2: Use Trigonometry

Using the tangent function:

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

Find $h_A$:

$\tan(32^\circ) = \frac{h_A}{20}$

$h_A = 20 \times \tan(32^\circ)$

$h_A \approx 12.5 \text{ meters}$

Total building height:

$H_A = h_A + 1.62$

$H_A = 12.5 + 1.62 = 14.12 \text{ meters}$

Find $h_B$:

$\tan(43^\circ) = \frac{h_B}{20}$

$h_B = 20 \times \tan(43^\circ)$

$h_B \approx 18.7 \text{ meters}$

Total height to the top of the antenna:

$H_B = h_B + 1.62$

$H_B = 18.7 + 1.62 = 20.32 \text{ meters}$

Step 3: Compute Antenna Height

$\text{Antenna height} = H_B - H_A$

$= 20.32 - 14.12$

$= 6.2 \text{ meters}$

Final Answer:

The antenna is 6 meters tall (rounded to the nearest meter).

---

Would you like a more detailed explanation or a diagram?

Related Questions:

1. How would the answer change if the distance from Isaiah to the building was different?
2. What if Isaiah's eye level was different?
3. How can we verify the calculations using another trigonometric function?
4. How would the problem change if the antenna was on a sloped roof?
5. How do we solve a similar problem when given different angles?

Tip:

Always double-check your tangent values to ensure accuracy when solving height-related problems!