The actual angle of inclination would be larger than the wrong angle, this is how we get it,
The airplane is flying at 250 mph, which is
[tex]\begin{gathered} \frac{250\times5280}{3600}= \\ 366.7\text{ft/s} \end{gathered}[/tex]When the airplane is at B, it sends sound to the observer on the ground, this sound travels a distance of x,
let's find x,
[tex]\begin{gathered} x=\frac{2500}{\sin 30} \\ x=5000ft \end{gathered}[/tex]Lets find d, the horizontal distance of the plane initially,
[tex]\begin{gathered} d=5000\cos 30 \\ d=4330ft \end{gathered}[/tex]For the 5000ft the sound traveled, it took some time, which is,
[tex]t=\frac{dis\tan ce}{\text{speed}}=\frac{5000}{1100}=4.55\text{seconds}[/tex]The plane is moving at 366.7ft/s, in 4.55 seconds it would be at point A, it would have covered a distance equivalent to d - c , this distance is also equal to the airplane's speed times time(4.55 seconds), we have:
[tex]\begin{gathered} d-c=366.7\times4.55 \\ d-c=1666.8ft \end{gathered}[/tex]But d= 4330ft , so:
[tex]\begin{gathered} c=4330-1666.8 \\ c=2663.18ft\text{.} \end{gathered}[/tex]So, our unknown angle can be gotten from trigonometrical relations,
[tex]\begin{gathered} \tan y=\frac{2500}{2663.18} \\ \tan y=0.9387 \\ y=\tan ^{-1}0.9387 \\ y=43.18^o\approx43^o \end{gathered}[/tex]So, you should look up at an angle of 43 degrees, to spot the airplane.
