Answer :
- Calculate the height from Taub's eyes to point A using the tangent function: $height_{A,eyes} = 20 \times tan(43^{\circ})$.
- Calculate the height from Taub's eyes to point B using the tangent function: $height_{B,eyes} = 20 \times tan(47^{\circ})$.
- Determine the height of the antenna by subtracting the two heights: $height_{antenna} = height_{B,eyes} - height_{A,eyes}$.
- Round the result to the nearest meter: $\boxed{3}$ meters.
### Explanation
1. Analyze the problem and given data
First, we need to find the height from the ground to point A (the roof) and point B (the top of the antenna). We know Taub is standing 20 meters away from the building, and her eyes are 1.57 meters above the ground. The angles of elevation to the roof and the top of the antenna are 43 and 47 degrees, respectively.
2. Calculate height from eyes to point A
Let's calculate the height from Taub's eyes to point A. We can use the tangent function: $tan(43^{\circ}) = \frac{height_{A,eyes}}{20}$. Therefore, $height_{A,eyes} = 20 \times tan(43^{\circ})$.
3. Calculate height from eyes to point B
Similarly, let's calculate the height from Taub's eyes to point B. Again, we use the tangent function: $tan(47^{\circ}) = \frac{height_{B,eyes}}{20}$. Therefore, $height_{B,eyes} = 20 \times tan(47^{\circ})$.
4. Calculate the height of the antenna
Now, we calculate the height of the antenna, which is the difference between the height from Taub's eyes to point B and the height from Taub's eyes to point A: $height_{antenna} = height_{B,eyes} - height_{A,eyes} = 20 \times tan(47^{\circ}) - 20 \times tan(43^{\circ})$. The result of this operation is approximately 2.797 meters.
5. Round the result
Finally, we round the height of the antenna to the nearest meter: $height_{antenna} \approx 3$ meters.
6. State the final answer
The height of the antenna is approximately 3 meters.
### Examples
Understanding angles of elevation and trigonometric functions like tangent is useful in various real-world scenarios, such as surveying land, determining the height of buildings or mountains, and even in navigation. For instance, sailors use sextants to measure the angle of elevation of stars to determine their position at sea. Similarly, construction workers use theodolites to measure angles and ensure structures are built correctly.
- Calculate the height from Taub's eyes to point B using the tangent function: $height_{B,eyes} = 20 \times tan(47^{\circ})$.
- Determine the height of the antenna by subtracting the two heights: $height_{antenna} = height_{B,eyes} - height_{A,eyes}$.
- Round the result to the nearest meter: $\boxed{3}$ meters.
### Explanation
1. Analyze the problem and given data
First, we need to find the height from the ground to point A (the roof) and point B (the top of the antenna). We know Taub is standing 20 meters away from the building, and her eyes are 1.57 meters above the ground. The angles of elevation to the roof and the top of the antenna are 43 and 47 degrees, respectively.
2. Calculate height from eyes to point A
Let's calculate the height from Taub's eyes to point A. We can use the tangent function: $tan(43^{\circ}) = \frac{height_{A,eyes}}{20}$. Therefore, $height_{A,eyes} = 20 \times tan(43^{\circ})$.
3. Calculate height from eyes to point B
Similarly, let's calculate the height from Taub's eyes to point B. Again, we use the tangent function: $tan(47^{\circ}) = \frac{height_{B,eyes}}{20}$. Therefore, $height_{B,eyes} = 20 \times tan(47^{\circ})$.
4. Calculate the height of the antenna
Now, we calculate the height of the antenna, which is the difference between the height from Taub's eyes to point B and the height from Taub's eyes to point A: $height_{antenna} = height_{B,eyes} - height_{A,eyes} = 20 \times tan(47^{\circ}) - 20 \times tan(43^{\circ})$. The result of this operation is approximately 2.797 meters.
5. Round the result
Finally, we round the height of the antenna to the nearest meter: $height_{antenna} \approx 3$ meters.
6. State the final answer
The height of the antenna is approximately 3 meters.
### Examples
Understanding angles of elevation and trigonometric functions like tangent is useful in various real-world scenarios, such as surveying land, determining the height of buildings or mountains, and even in navigation. For instance, sailors use sextants to measure the angle of elevation of stars to determine their position at sea. Similarly, construction workers use theodolites to measure angles and ensure structures are built correctly.
