Question

Two sensors are spaced 700 feet apart along the approach to a small airport. when an aircraft is nearing the airport, the angle of elevation from the first sensor to the aircraft is 20 degrees, and from the second sensor to the aircraft it is 12 degrees. determine the height of the aircraft in feet at this time?

Answer 1

I attached the sketch of this problem.
One thing that we can notice right away is that:

`h=c_1\sin(\theta)=c_2\sin(\alpha)`
The second thing we can notice is that:

`l=c_1\cos(\theta)-c_2\cos(\alpha)`
This is basic trigonometry. You can always think of cosine as a projection on the x-axis and sine as the projection on the y-axis. From these two equations, we can find c_1 or c_2 and find the height.

`c_1\sin(\theta)=c_2\sin(\alpha)\\ l=c_1\cos(\theta)-c_2\cos(\alpha)\\ c_2=(c_1\sin(\theta))/(\sin(\alpha))\\ l=c_1\cos(\theta)-c_1(\sin(\theta))/(\sin(\alpha)) \cos(\alpha)\\ l=c_1\cos(\theta)-c_1(\sin(\theta))/(\tan(\alpha))\\ l=c_1(\cos(\theta)-(\sin(\theta))/(\tan(\alpha)))\\ c_1=l\cdot (\cos(\theta)-(\sin(\theta))/(\tan(\alpha)))^(-1)=1720.26$ft`
Now to find the height we use the first equation:

`h=c_1\sin(\theta)=357.66$ft`