SOLUTION: From a window in an apartment building, the angle of depressions to parked car is 24 degrees. From a window 15m lower the angle of depression is 19 degrees. How far is the car from

Algebra ->  Trigonometry-basics -> SOLUTION: From a window in an apartment building, the angle of depressions to parked car is 24 degrees. From a window 15m lower the angle of depression is 19 degrees. How far is the car from      Log On


   

Question 1151672: From a window in an apartment building, the angle of depressions to parked car is 24 degrees. From a window 15m lower the angle of depression is 19 degrees. How far is the car from the foot of the building?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Diagram

Figure 1 shows the original diagram with the following points defined
Point A = base of the building
Point B = window 15 meters below point C (this has angle of depression 19 degrees)
Point C = first, and uppermost, window mentioned (that has angle of depression of 24 degrees)
Point D = car's location
Point E = used to help form the angle of depression 19 degrees
Point F = used to help form the angle of depression 24 degrees

Side Lengths:
AB = y
AD = x
BC = 15
AC = AB+BC = y+15

Angles of depression (marked in blue):
Angle FCD = 24 degrees
Angle EBD = 19 degrees
Angles complementary to the blue angles (marked in red)
Angle DCA = 66 degrees
Angle DBA = 71 degrees
Each adjacent red and blue angle add to 90 degrees.

Figure 2 represents triangle ABD separated from triangle ACD

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Use the tangent ratio. Focus on triangle ABD in figure 2 (ignore point E)
tan(angle) = opposite/adjacent
tan(B) = AD/AB
tan(71) = x/y
y*tan(71) = x
x = y*tan(71) We will use this equation later.

Set up another tangent ratio. Focus on triangle ACD in figure 2 (ignore point F)
tan(angle) = opposite/adjacent
tan(C) = AD/AC
tan(66) = x/(15+y)
tan(66)*(15+y) = x
tan(66)*15+tan(66)*y = x
15*tan(66)+y*tan(66) = x
15*tan(66)+y*tan(66) = y*tan(71) ..... plug in x = y*tan(71)
15*tan(66) = y*tan(71)-y*tan(66)
15*tan(66) = y*(tan(71)-tan(66))
15*tan(66)/(tan(71)-tan(66)) = y
51.1879021303067 = y
y = 51.1879021303067

Use that y value to find x
x = y*tan(71)
x = 51.1879021303067*tan(71)
x = 148.660462172242
x = 148.66

Answer: The car is approximately 148.66 meters away from the base of the building.