Alice and Bob live $20$ miles apart. Alice looks due north and sees an airplane; Bob looks due west and sees the same airplane. The angle of elevation is $30^\circ$ from Alice and $45^\circ$ from Bob. What is the airplane's altitude, in miles?

1. Let $P$ be the point on the ground below the airplane and $h$ its altitude. Then $AP=h\cot\alpha$ (due north of Alice) and $BP=h\cot\beta$ (due west of Bob), and $AP\perp BP$, so $d^2=AB^2=h^2\cot^2\alpha+h^2\cot^2\beta$.
2. With $d=20,\alpha=30^\circ,\beta=45^\circ$: $\cot30^\circ=\sqrt3,\cot45^\circ=1$, so $400=h^2(3+1)=4h^2$.
3. $h=\boxed{10}$.