Question

An aircraft is spotted by two observers who are 1000 feet apart. As the airplane passes over the line joining them, each observer takes a sighting of the angle of elevation to the plane, as indicated in the figure. How high is the airplane?

Answer 1

The airplane is 335.17 feet high.

To solve this problem, we can use trigonometry.

We know that the distance between the two observers is 1000 feet. We also know the angles of elevation to the plane from each observer: 50 degrees and 35 degrees.

We want to find the height of the airplane, which is represented by the line segment h in the diagram.

To do this, we can use the following trigonometric equation:

tan(theta) = height / distance

where `theta` is the angle of elevation and `distance` is the distance between the observer and the plane.

We can use this equation to write down two equations for the two observers:

tan(50 degrees) = h / 1000 feet

tan(35 degrees) = h / (1000 feet - h)

Now we have two independent equations, and we can solve for the two unknowns: `h` and `theta`.

One way to solve for `h` is to use the following steps:

1. Solve the second equation for `h` in terms of `theta`:

h = 1000 feet * tan(35 degrees) / (1 - tan(35 degrees))

2. Substitute this expression for `h` in the first equation:

tan(50 degrees) = (1000 feet * tan(35 degrees) / (1 - tan(35 degrees))) / 1000 feet

3. Simplify this equation and solve for `theta`:

tan(50 degrees) = tan(35 degrees) / (1 - tan(35 degrees))

theta = 50 degrees

4. Now that we know `theta`, we can substitute it back into the second equation to solve for `h`:

h = 1000 feet * tan(35 degrees) / (1 - tan(35 degrees))

h = 335.17 feet

Therefore, the height of the airplane is 335.17 feet.

Answer 2

see the attached figure to better understand the problem

we know that
PB+BQ=1000 ft------->PB=1000-BQ-------> equation 1

in the triangle PAB
tan 50=AB/PB----------> AB=PB*tan 50------> equation 2

in the triangle ABQ
tan 25=AB/BQ------> AB=BQ*tan 25-------> equation 3

equals equation 2 and equation 3
PB*tan 50=BQ*tan 25--------> equation 4

substitute equation 1 in equation 4

[1000-BQ]*tan 50=BQ*tan 25-----> 1000*tan 50-BQ*tan 50=BQ*tan 25
BQ*[tan 25+tan 50]=1000*tan 50-----> BQ=1000*tan 50/[tan 25+tan 50]
BQ=718.76 ft
PB=1000-718.76-----> PB=281.24 ft
AB=PB*tan 50-----> 281.24*tan 50------> 335.17 ft

the answer is
335.17 ft