Question 1187929: From the top of Joe’s grocery store, the angle of depression to the base of the adjacent store is 32 degrees. If the distance between the two buildings is 20 m, find the height of Joe’s store.
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! here's a reference on angle of depression.
https://www.varsitytutors.com/hotmath/hotmath_help/topics/angles-of-elevation-and-depression
here's my diagram of your problem.
line AB is a horizontal line from the top of joe's store to the top of the adjacent store.
in the diagram, that is the line BC.
if the adjacent store is not the same height as joe's store, it doesn't matter.
AB is still a horizontal line that intersects with the vertical line that is the vertical side of the adjacent store, or an extension of the vertical side of the adjacent store.
further diagrams down below will show you what i mean.
you have a rectangle that is formed.
that rectangle is ABCD.
it is formed by the vertical side of joe's store and the vertical line that represents the vertical side of the adjacent store.
if you draw a diagonal from the top of joe's store to the bottom of the adjacent store, you get two triangles.
they are triangle ABC and ADC.
angle BAC is the angle of depression.
this is called angle A.
angle A is the angle of depression.
angle ACD is an angle of elevation.
this is called angle C.
angle C is called the angle of elevation.
the length of side AD is equal to x.
the length of side BC is also equal to x.
tan(A) = BC / AB.
since angle A = 32 degrees and BC = x and AB = 20, you get:
tan(32) = x/20
solve for x to get:
x = 20 * tan(32) = 12.49738704.
x is the length of side BC.
tan(C) = AD / DC.
since C = 32 degrees and AD = x and DC = 20, you get:
tan(32) = x/20
solve for x to get:
x = 20 * tan(32) = 12.49738704
the length of the line BCis the same as the length of the line AD.
AD is the height of joe's store.
your solution is that the height of joe's store is 12.49738704 meters.
the height of the adjacent bulding is not important, as you will see in the following diagram.
in the top diagram, the adjacent store is not as high as joe's store.
you just extend the vertical line BC so that it meets with the horizontal line AB and form your rectangle ABCD.
in the bottom diagram, the adjacent store is higher than joe's store.
you just intersect your horizontal AB with the part of the vertical height of the adjacent store which is represented by the vertical line BC so that it forms your rectangle ABCD.
the height that you are looking for is the height of joe's store which is the length of line AD.
since you formed a rectangle, the horizontal line AB is equal to and parallel with the the horizontal line DC, and the vertical line AD is equal to and parallel with the vertical line BC.
angle A is equal to angle C because they are alternate interior angles of the parallel lines AB and CD.
let me know if you have any questions.
theo
Answer by ikleyn(52803)  (Show Source):
You can put this solution on YOUR website! .
Hello,
the solution to this problem is, actually, in ONE LINE
the height of the Joe's store is 20*tan(32°) = 20*0.6249 = 12.498 m = 12.5 meters (rounded). ANSWER
Solved.
|
|
|
