SOLUTION: I'm having trouble figuring this out. The angles of elevation to an airplane from two points A and B on level ground are 55° and 72°, respectively. The points A and B are 2.8 m

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Question 1206616: I'm having trouble figuring this out.
The angles of elevation to an airplane from two points A and B on level ground are 55° and 72°, respectively. The points A and B are 2.8 miles apart, and the airplane is east of both points in the same vertical plane.
a) Find the distance between the plane and point B. (Round your answer to two decimal places.)
(b) Find the altitude of the plane. (Round your answer to two decimal places.)
(c) Find the distance the plane must travel before it is directly above point A. (Round your answer to two decimal places.)
Found 2 solutions by Edwin McCravy, math_tutor2020:
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!



system%28tan%2853%5Eo%29=h%2F%282.8%2Bx%29%2C+tan%2872%5Eo%29=h%2Fx%29



system%28%282.8%2Bx%29tan%2853%5Eo%29=h%2C+x%2Atan%2872%5Eo%29=h%29

%282.8%2Bx%29tan%2853%5Eo%29=+x%2Atan%2872%5Eo%29%29

2.8tan%2853%5Eo%29%2Bx%2Atan%2853%5Eo%29=+x%2Atan%2872%5Eo%29%29

2.8tan%2853%5Eo%29=x%2Atan%2872%5Eo%29-x%2Atan%2853%5Eo%29%29

2.8tan%2853%5Eo%29=x%28tan%2872%5Eo%29-tan%2853%5Eo%29%5E%22%22%29%29

%282.8tan%2853%5Eo%29%29%2F%28tan%2872%5Eo%29-tan%2853%5Eo%29%29=x%29

2.122497045=x%29

Substitute in

x%2Atan%2872%5Eo%29=h

2.122497045%2Atan%2872%5Eo%29=h

6.532374213=h  <--altitude of plane [answer to (b)]

x%2F%28PB%29=cos%2872%5Eo%29

x=PB%2Acos%2872%5Eo%29

x%2Fcos%2872%5Eo%29=PB

2.122497045%2Fcos%2872%5Eo%29=PB

6.86854472=PB <--distance between the plane and point B [answer to (a)]

Find (c) all by yourself, and round off as you were told.

Edwin

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

I'll focus on part (b) only.

Review this similar question
https://www.algebra.com/algebra/homework/Triangles/Triangles.faq.question.1199063.html
On that solution page I provide the derivation of this formula
h = d*u*v/(u-v)
where,
h = height or altitude of the object
d = distance between observation points
u = tangent of larger angle
v = tangent of smaller angle

In this case,
d = 2.8
u = tan(72)
v = tan(55)

So we can quickly determine the altitude to be:
h = d*u*v/(u-v)
h = 2.8*tan(72)*tan(55)/( tan(72) - tan(55) )
h = 7.46 miles approximately
Make sure your calculator is in degrees mode.

Tutor Edwin has the right idea, but I'm not sure where he got the 53 degree angle from. It should be 55 instead.

Extra info:
Multiply by 5280 to convert from miles to feet.
7.46 miles = 7.46*5280 = 39388.8 feet