SOLUTION: a pilot flying horizontally at a constant height, measures the angle of depression to a landmark to be 18 degrees. After flying 700m, the new angle of depression to the landmark is

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Question 1151609: a pilot flying horizontally at a constant height, measures the angle of depression to a landmark to be 18 degrees. After flying 700m, the new angle of depression to the landmark is 21 degrees. At what height is the pilot flying.
Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website!

There are at least two methods to solve this problem.

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Method 1

Construct the diagram to look something like this. The goal is to find h

The diagram is not to scale.
Point A = plane's starting position
Point B = plane's position after flying 700 meters
Point C = landmark on the ground
Point D = used to help form the angle of depression of 21 degrees
Point E = point on ground directly under point A
Point F = point on ground directly under point B
Each pair of adjacent red and blue angles add up to 90 degrees (ie they are complementary angles)

The diagram may seem a bit cluttered, so you can peel triangle AEC and triangle BFC apart to get this

Focus on triangle BFC
tan(angle) = opposite/adjacent
tan(B) = FC/FB
tan(69) = x/h
h*tan(69) = x
x = h*tan(69)

Move onto triangle AEC
tan(angle) = opposite/adjacent
tan(A) = EC/AE
tan(72) = (700+x)/h
tan(72) = (700+h*tan(69))/h ... plug in x = h*tan(69)
h*tan(72) = 700+h*tan(69)
h*tan(72)-h*tan(69) = 700
h*(tan(72)-tan(69)) = 700
h = 700/(tan(72)-tan(69))
h = 1481.18533068004 is the approximate height in meters.

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Method 2

The diagram will be similar, but it now looks like this

The diagram is not to scale.
Points A,B,C,D are defined the same way as in the previous diagram.
Angle B = 159 because angle ABC = 180-(angle DBC) = 180-21 = 159
Angle C = 3 degrees comes from the fact that A+B+C = 180

Focus solely on triangle ABC (ie ignore point D)
Use the law of sines to find side 'a'.
a/sin(A) = c/sin(C)
a/sin(18) = 700/sin(3)
a = sin(18)*700/sin(3)
a = 4133.14118229429

Now we can use the SAS triangle area formula
area = (1/2)*side1*side2*sin(included angle)
area = (1/2)*a*c*sin(B)
area = (1/2)*4133.14118229429*700*sin(159)
area = 518414.865738014
This is the approximate area of triangle ABC.

Finally, turn to the formula
area = (1/2)*base*height
to help find the height of the triangle ABC

area = (1/2)*base*height
518414.865738014 = (1/2)*700*h
518414.865738014*2 = 700*h
1036829.73147602 = 700*h
700*h = 1036829.73147602
h = 1036829.73147602/700
h = 1481.18533068002 which was roughly the same approximate height we got earlier.