Related rates, my answer differs from the book, misprint or me?

Question

Did my answer go wrong or does the book have a misprint?(there have been some inconstancies between the definitions used in the chapters and answer key, like two different authors, though only one is listed)

The problem: An airplane is flying 500 miles per hour horizontal one mile high over a radar station. Find the rate at which the distance is increasing when the plane is 2 miles from the station.

My answer is $1000/\sqrt5$ or $200*\sqrt5$

The book gives $250*\sqrt3$

My method: triangle abc, a=1, b=2, and c is the hypotenuse, $db/dt=500$, and $c^2=1^2+b^2$, and I want $dc/dt$ at b=2

I took the derivative: $2c*\frac{dc}{dt}=0+2b*\frac{db}{dt}$,

solved for $dc/dt$; $dc/dt=\frac{2b*db/dt}{2c}$

Substitute the variables; $c=\sqrt{1+4}$ and so $\frac{dc}{dt}=\frac{2*2*500}{2*\sqrt5}=\frac{1000}{\sqrt5}$

Answer 1

It may be a misprint, but I don't think your answer is right either. The question asks for $dc/dt$ when the distance from the station is 2 - this means when c=2 not when b=2. Other than that your answer is correct; plugging a=1, b=$\sqrt 3$, c=2 in, I get $250\sqrt3$ (not $250/\sqrt3$)

Answer 2

You approach is good, you just mixed up $b$ and $c$. When plane is two miles from the station, $c=2$, $b=\sqrt{3}$. So I think the answer should be $250\sqrt{3}$.