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 85.0 m. In part a the person is looking at the base of the antenna, and his line of sight makes an angle of 35.0° 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 38.0° with the horizontal. How tall is the antenna?
If the building's height is b, then
tan35° = b/85
Now, knowiong b, you can find the antenna's height a, using
tan38° = (b+a)/85
Or, I guess you could solve for a, giving
a = 85(tan38° - tan35°)
To find the height of the antenna, we can use trigonometry. Let's consider the triangle formed by the person's line of sight, the horizontal distance to the building, and the height of the antenna.
In part a, the person is looking at the base of the antenna, forming a right triangle with the horizontal distance of 85.0 m. We can define the base of the triangle as 85.0 m.
Using the given angle of 35.0°, we can label one of the sides adjacent to the angle as the height of the antenna (h) and the other side as the opposite side of the angle.
Using the trigonometric function tangent (tan), we can set up the following equation:
tan(35.0°) = h / 85.0
To find the height (h), we can rearrange the equation:
h = 85.0 * tan(35.0°)
Now we can calculate the height of the antenna.
h = 85.0 * tan(35.0°)
h ≈ 85.0 * 0.7002
h ≈ 59.54 m
In part b, we can follow the same approach. The person is now looking at the top of the antenna, forming a right triangle with the same horizontal distance of 85.0 m.
Using the given angle of 38.0°, we can label the side adjacent to the angle as the height of the antenna (h), and the other side as the opposite side of the angle.
Using the trigonometric function tangent (tan), we can set up the following equation:
tan(38.0°) = h / 85.0
To find the height (h), we can rearrange the equation:
h = 85.0 * tan(38.0°)
Now let's calculate the height of the antenna using this equation.
h = 85.0 * tan(38.0°)
h ≈ 85.0 * 0.8080
h ≈ 68.68 m
Therefore, the height of the antenna is approximately 68.68 meters.
To find the height of the antenna, we can use trigonometry and set up two right triangles.
Let's start with part a:
In part a, the person is looking at the base of the antenna, forming a right triangle with the horizontal line. The angle of elevation is 35.0°.
Let's label the height of the antenna as h.
Using trigonometry, we can set up the equation:
tan(35.0°) = h / 85.0m
Now, let's solve for h:
h = 85.0m * tan(35.0°)
h ≈ 85.0m * 0.7002
h ≈ 59.51m
So, the height of the antenna in part a is approximately 59.51 meters.
Now let's move on to part b:
In part b, the person is looking at the top of the antenna, forming another right triangle with the horizontal line. The angle of elevation is 38.0°.
Using trigonometry, we can set up the equation:
tan(38.0°) = (h + x) / 85.0m
Here, x represents the additional distance from the base of the antenna to the top.
Rearranging the equation:
(h + x) = 85.0m * tan(38.0°)
Now, solve for h + x:
(h + x) ≈ 85.0m * 0.7813
(h + x) ≈ 66.56m
Since we already know that the height of the antenna is h = 59.51m, we can find x by subtracting h:
x = (h + x) - h
x ≈ 66.56m - 59.51m
x ≈ 7.05m
So, the additional distance from the base to the top of the antenna is approximately 7.05 meters.
Therefore, the total height of the antenna is the sum of h and x:
Total height = h + x
Total height ≈ 59.51m + 7.05m
Total height ≈ 66.56m
Thus, the height of the antenna is approximately 66.56 meters.