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Caleb is trying to find the height of a radio antenna on the roof of a local building. He stands at a horizontal distance of 17 meters from the building. The angle of elevation from his eyes to the roof (point \( A \) ) is \( 38^{\circ} \), and the angle of elevation from his eyes to the top of the antenna (point \( B \) ) is \( 53^{\circ} \). If his eyes are 1.58 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.

Ask by Burns Hall. in the United States
Feb 03,2025

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The height of the antenna is approximately 9 meters.

Solución

To determine the height of the antenna (\( AB \)), we can use trigonometric relationships based on the information provided. 1. **Understand the Scenario:** - **Horizontal Distance (\( x \))**: 17 meters from Caleb to the building. - **Eye Level Height (\( y_0 \))**: 1.58 meters from the ground. - **Angles of Elevation**: - To the roof (\( A \)): \( 38^\circ \) - To the top of the antenna (\( B \)): \( 53^\circ \) 2. **Calculate the Vertical Heights from Eye Level:** - **Height to Roof (\( y_A \))**: \[ y_A = x \cdot \tan(38^\circ) \] - **Height to Antenna (\( y_B \))**: \[ y_B = x \cdot \tan(53^\circ) \] 3. **Plug in the Values:** - Using a calculator: \[ \tan(38^\circ) \approx 0.7813 \] \[ \tan(53^\circ) \approx 1.3270 \] - Calculate each height: \[ y_A = 17 \times 0.7813 \approx 13.281 \text{ meters} \] \[ y_B = 17 \times 1.3270 \approx 22.559 \text{ meters} \] 4. **Determine the Height of the Antenna (\( AB \)):** \[ AB = y_B - y_A \approx 22.559 - 13.281 = 9.278 \text{ meters} \] 5. **Final Answer:** Rounding to the nearest meter: \[ AB \approx 9 \text{ meters} \] **Therefore, the height of the antenna is approximately **9 **meters.**

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To find the height of the radio antenna, we can use some trigonometric principles! First, we need to calculate the height of point \( A \) (the roof) using the angle of elevation to point \( A \). Using the tangent function: \[ h_A = d \cdot \tan(\theta_A) + h_{eyes} \] where \( d = 17 \) meters, \( \theta_A = 38^{\circ} \), and \( h_{eyes} = 1.58 \) meters. \[ h_A = 17 \cdot \tan(38^{\circ}) + 1.58 \] Calculating this gives approximately \( h_A \approx 17 \cdot 0.7813 + 1.58 \approx 16.78 + 1.58 \approx 18.36 \) meters. Now, for point \( B \) (the top of the antenna): \[ h_B = d \cdot \tan(\theta_B) + h_{eyes} \] where \( \theta_B = 53^{\circ} \). \[ h_B = 17 \cdot \tan(53^{\circ}) + 1.58 \] Calculating this gives approximately \( h_B \approx 17 \cdot 1.3270 + 1.58 \approx 22.55 + 1.58 \approx 24.13 \) meters. Finally, to find the height of the antenna (the distance from point \( A \) to point \( B \)): \[ h_{antenna} = h_B - h_A = 24.13 - 18.36 \approx 5.77 \] Rounding this to the nearest meter, the height of the antenna is approximately 6 meters!

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