SOLUTION: There is an antenna on the top of a building. From a location 400 feet from the base of the building, the angle of elevation to the top of the building is measured to be 41°. From
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Question 1203325: There is an antenna on the top of a building. From a location 400 feet from the base of the building, the angle of elevation to the top of the building is measured to be 41°. From the same location, the angle of elevation to the top of the antenna is measured to be 44°. Find the height of the antenna. (Round your answer to three decimal places.) Answer by math_tutor2020(3817) (Show Source):
Let's start off with a rough sketch of the diagram
The key points are
A = bottom of the person's feet
B = location of the person's eye
C = base of the building
D = point directly across from B and directly above point C
E = top of the building
F = top of the antenna
We have these variables
x = distance from E to F = height of the antenna
y = distance from C to D
I've marked angle DBE = 41 in green, and angle DBF = 44 in blue.
Focus on triangle DEB.
tan(angle) = opposite/adjacent
tan(angle DBE) = DE/BD
tan(41) = y/400
y = 400*tan(41)
This will be used in a substitution step in the next section.
Now focus on triangle DBF
tan(angle) = opposite/adjacent
tan(angle DBF) = FD/BD
tan(44) = (FE+ED)/BD
tan(44) = (x+y)/400
tan(44) = (x+400*tan(41))/400 ....... substitution
400*tan(44) = x+400*tan(41)
x = 400*tan(44)-400*tan(41)
x = 400*( tan(44)-tan(41) )
x = 38.560814796339
x = 38.561
This value is approximate.
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