The drawing shows a person looking at a building on top of which an antenna is mounted. The horizontal distance between the person's eyes and the building is L = 81.0 m. In part (a) the person is looking at the base of the antenna, and his line of sight makes an angle of èa = 36.7° with the horizontal. In part (b) the person is looking at the top of the antenna, and his line of sight makes an angle of èb = 39° with the horizontal. How tall is the antenna?
If the vertical distances to the bottom and top of the antenna are a and b, then
a/81 = tan 36.7°
b/81 = tan 39°
the height of the antenna is b-a
To determine the height of the antenna, we can use trigonometry and the given angles.
Let's start with Part (a), where the person is looking at the base of the antenna. We have the following information:
- The horizontal distance between the person's eyes and the building is L = 81.0 m.
- The angle between the person's line of sight and the horizontal is èa = 36.7°.
We can consider a right triangle formed by the person, the base of the antenna, and a point on the ground directly below the person's eyes. The height of the antenna will be the length of the side opposite to the angle èa.
Using trigonometry, we can use the tangent function to calculate the height (h) of the antenna:
tan(èa) = h / L
Rearranging the equation, we get:
h = L * tan(èa)
Substituting the known values, we have:
h = 81.0 m * tan(36.7°)
Calculating this value, we find that the height of the antenna in Part (a) is approximately 62.4 meters.
Moving on to Part (b), where the person is looking at the top of the antenna, we have the following information:
- The horizontal distance between the person's eyes and the building is still L = 81.0 m.
- The angle between the person's line of sight and the horizontal is èb = 39°.
Similar to Part (a), we can use trigonometry and the tangent function to calculate the height (h) of the antenna:
h = L * tan(èb)
Substituting the known values, we have:
h = 81.0 m * tan(39°)
Calculating this value, we find that the height of the antenna in Part (b) is approximately 71.2 meters.
In conclusion, the antenna is approximately 62.4 meters tall when the person is looking at the base (Part a) and approximately 71.2 meters tall when the person is looking at the top (Part b).