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?
85 (tan 38 - tan 35) meters
This is more of a trigonometry than a physics question.
the question is a bit ambiguous in not specifying whether line of sight is UP from the horizontal or DOWN. the correct answer could be:
h = 85m[tan 38deg + tan 35deg]
...if the observer is in a window of a nearby building or on a hilltop, etc.
[many professors are ambiguous because they know the answer THEY want.]
To find the height of the antenna, we can use trigonometry and set up a proportion. Let's first find the height of the building using the given information in part a.
In part a, the person is looking at the base of the antenna, forming an angle of 35.0 degrees with the horizontal. We can assume the person's eyes are at the same height as the base of the building.
Let's define the following variables:
h = height of the building
θ = angle of the line of sight with the horizontal (35.0 degrees)
d = horizontal distance between the person's eyes and the building (85.0 m)
Using trigonometry, we know that:
tan(θ) = h/d
Rearranging the equation to solve for h, we get:
h = d * tan(θ)
Substituting the given values:
h = 85.0 m * tan(35.0 degrees)
Using a calculator:
h = 48.75 m
Now, let's find the height of the antenna using the given information in part b.
In part b, the person is looking at the top of the antenna, forming an angle of 38.0 degrees with the horizontal.
Using the same trigonometric relationship as before, we can set up the following proportion:
h (height of the building) + height of the antenna) / d = tan(θ)
Let's define another variable:
a = height of the antenna
Now we can rearrange the equation to solve for the height of the antenna:
a = (tan(θ) * d) - h
Substituting the given values:
a = tan(38.0 degrees) * 85.0 m - 48.75 m
Using a calculator:
a = 76.27 m
Therefore, the height of the antenna is approximately 76.27 meters.
To find the height of the antenna, we can use trigonometric ratios such as tangent.
First, let's consider part a where the person is looking at the base of the antenna at an angle of 35.0 degrees with the horizontal.
We have the horizontal distance between the person's eyes and the building, which is 85.0 m. Let's call this distance "d".
Now, we can use the tangent ratio to find the height of the antenna.
The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the antenna, and the adjacent side is the horizontal distance between the person's eyes and the building.
So, tan(35.0) = height of the antenna / 85.0 m
To find the height of the antenna, we can rearrange the equation:
height of the antenna = 85.0 m * tan(35.0)
Calculating this, we find that the height of the antenna in part a is approximately 66.08 m.
Now, let's move on to part b where the person is looking at the top of the antenna at an angle of 38.0 degrees with the horizontal.
Using the same approach, we can set up another equation:
height of the antenna = 85.0 m * tan(38.0)
Calculating this, we find that the height of the antenna in part b is approximately 77.18 m.
Therefore, the height of the antenna is approximately 77.18 m.