For math requirements

Question

The tower is 500 meters high. Suppose a building is erected such that the base of the building is on the same plane as the base of the tower, the angle of elevation from the top of the building to the top of the tower is 75.24 and then angle of the depression from the top of the building to the foot of the tower is 55.05. How high would the building have to be?

Answer 1

For clarity, since the angle of elevation is given from the top of the building to the top of the tower,

that implies the tower is TALLER than the building. Building on your left, tower on your right.

Draws the extended line from the top of the building parallel to the ground, perpendicular to the tower.

Labels the top right point of the building T.

Labels the intersection of the line just drawn to where it intersects the tower, point N.

That is line TN extends from the top right point of the building , parallel to the ground, and

intersects the tower at point N.

Labels point M at the top of the tower.

Labels point A the bottom right corner of the building.

Labels point B the base of the tower.

LIne segment MB is given to be 500.

Let AB = x and TA = height of building = y

That is, the distance from the base of the building to the tower is x.

Let angle Theta be the angle of elevation from the top of the building to

the top of the tower. That it, theta = angle MTN = 75.24 as GIVEN.

Angle TNB = angle = TNM = 90

Let angle N = angle TMN = 90 - 75.24 = 14.76

Let angle alpha = angle NTB be the given angle of depression

from the top of the building to the base of the tower. That is, alpha = 55.05

THen, since TN is parallel to AB, angle TBA is also equal to alpha=55.05.

This forces angle epsilon = angle TBN = 34.95 = angle ATB by parallel lines.

That is, angles alpha and epsilon are complementary.

tan theta = (500-y)/x

tan alpha = y/x ---> x * tan alpha = y

substitutes into the equation above in bold.

tan theta = (500 - x*tan alpha)/x

x * tan theta = 500 - x * tan alpha

x * tan theta + x * tan alpha = 500

x ( tan theta + tan alpha) = 500

x = 500 / ( tan theta + tan alpha) = 95.6685265+ <---- distance between building and tower

which forces y = 136.8828 <--- height of the building

Answer 2