A 12m tall antenna sits on top of a building. A person is standing some distance away from the building. If the angle of elevation between the person and the top of the antenna is 60°, and the angle of elevation between the person and the top of the building is 45°, how tall is the building and how far away is the person standing?
To find the height of the building and the distance to the person, we can use trigonometry. Let's call the height of the building h and the distance to the person x.
Using the given information, we can start by drawing a diagram:
B
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/ | h
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/______|
A x C
In triangle ABC, angle ABC is 45° and angle ACB is 60°.
First, we can find the height of the antenna using the tangent function. The tangent of an angle is equal to the ratio of the opposite side to the adjacent side.
In triangle ABC, the opposite side is h (height of the antenna) and the adjacent side is x (distance to the person).
So we have:
tan(60°) = h / x
To solve for h, we multiply both sides of the equation by x:
x * tan(60°) = h
Next, let's find the height of the building. In triangle ABC, the height of the building is equal to the height of the antenna plus the height of the person.
Using the tangent function again, we can find the height of the person. The tangent of an angle is equal to the ratio of the opposite side to the adjacent side.
In triangle ABC, the opposite side is h (height of the person) and the adjacent side is x (distance to the person). The angle is 45°.
So we have:
tan(45°) = h / x
To solve for h, we multiply both sides of the equation by x:
x * tan(45°) = h
Now we can substitute the values we previously found into the equation for the height of the building:
Building height = Antenna height + Person height
h_building = h + h_person
h_building = x * tan(60°) + x * tan(45°)
To find the distance x, we can use the fact that the height of the building is the same as the height of the antenna plus the height of the person:
h_building = x * tan(60°) + x * tan(45°)
From here, we can substitute the values and calculate the height of the building and the distance x.
Let
h = height of building
x = distance of observer from the building
Then, assuming that the antenna is not set back from the edge of the building, we have
1: h/x = tan45°
2: (h+12)/x = tan60°
From (1), h=x, so in (2),
(h+12)/h = √3
Now just find h.