Geometrical Question?

Question

Two surveyors where trying to find the difference in height between Pilgreen Place and Patterson Palace. They were able to get a few measurements, the angle of depression from the top of Pilgreen Place to the base of Patterson Palace is 37 degrees, also they found that Pilgreen Place is exactly 100m tall. Finally a key piece of information...the ANGLE OF DEPRESSION from the top of Patterson Palace to the top of Pilgreen Place is 55 degrees.
What is the difference in height between the two buildings?

>>Please explain answer briefly<<


http://www.e-zgeometry.com/pow/pictures/29.gif

Answer 1

Oh, thats easy stuff :)

tangent = opp / adj
cosine = adj / hyp
sine = opp / hyp

tangent(37 degrees) = 100 / adj
adj = 100 / tan(37 degrees)

This gives you the distance between the two buildings. Using this, you can find the height about the Pilgreen place.

tan(55) = height / adj
height = adj * tan (55)

This is the height difference they are interested in.

Answer 2

The drawing is missing the 55 degree angle from the top of Patterson to Pilgreen. In any event, first you need to calculate the distance between the two buildings. Since you have a right triangle with one angle of 37 degrees and an opposite leg of given length 100 m, you can find the distance x between the buildings by saying that tan(37) = 100/x ==> x = 100/tan(37). Now, knowing that oppsite interior angles of the transversal of parallel lines are congruent, we also have a right triangle with an angle of 55 degrees and an adjacent leg of x, which was just calculated. The unknown leg y is also the difference in height between the two buildings, and we can say that tan(55) = y/x ==> y = x*tan(55) = 100*tan(55)/tan(37), the final answer.

I uploaded an updated graphic showing where the legs x and y are. (See third reference.)

Answer 3

It's pretty simple actually, just that the sentence may be misleading. Here's a rough diagram:

Top of Patterson
l \
l \
l 55(\
l____Top of Pilgrim
l---------/l
l--------/ l
l------ / l100m
l----/)37 l
---/____ lbase
To find the difference in height, you would need to know the distance between the two buildings first. To do that, you can do this: 100tan37= 75.3554m.

Then, you can find out the height difference like this: 75.3554tan45=75.3554m. This means that the taller building is twice the height of the Pilgrim building.

I hope this helps. :)