SOLUTION: a pilot flying at 4000 ft approaches a runway and determines that the angles of depression to either end of the runway are 10 degrees and 6 degrees 40' find the length of the runwa

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Question 1110603: a pilot flying at 4000 ft approaches a runway and determines that the angles of depression to either end of the runway are 10 degrees and 6 degrees 40' find the length of the runway.
Found 2 solutions by Boreal, KMST:
Answer by Boreal(15235) About Me  (Show Source):
You can put this solution on YOUR website!
Draw this and there are right triangles, both with one leg 4000' long and the opposite angle the opposite to the depression or elevation to the plane.
Tangent of 10=4000/x
x=4000/tangent 10=22685 feet
x=4000/tangent 6.67=34222 feet
The runway is 11,537 feet long

Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
A sketch representing the situation is shown below.
For the a better illustration,
in the sketch the angles of depression are 30%5Eo and 60%5Eo .
runway=x-y ,
tan%28A%29=h%2Fx --> x=h%2Ftan%28A%29 ,
tan%28B%29=h%2Fy --> y=h%2Ftan%28B%29 .
With all length in feet,
h=4000 , A=6%5Eo%2240+%27%22=%286%262%2F3%29%5Eo=about+6.666667%5Eo , and B=10%5Eo ,
we can calculate a good approximate value of the runway length as
4000%2Ftan%286.666667%5Eo%29%22-%224000%2Ftan%2810%5Eo%29%22=%224000%2F0.116883%22-%224000%2F0.176327%22=%2234222-22685%22=%22%22about+11%2C500%22 .
The length of the runway is about highlight%28%2211%2C500+feet%22%29 .
We could carry more digits through the calculations,
and we could calculate the length of the runway as 11537feet ,
but considering the uncertainty of the altitude and angle measurements
a more precise result is not more meaningful.