Answer
$-600 \sqrt {5}\approx -1342$ mi/h
Work Step by Step
Reference to the right-angled triangle APB in the fig.
$AP=y=4 $ mi=Distance of anticraft missile from the point P
$ \frac{dy}{dt}=1200$ mi/h = Velocity of anticraft missile approaching the point P
$x=2$mi=Distance of the aircraft from the impact point P
$\frac{dx}{dt}=600$mi/h=Velocity of the aircraft towards the point P
Now
$z^2=x^2+y^2$ ....................... eq (1)
Putting $x=2$ miles, $y=4$miles in equation (1)
$z^2=2^2+4^2=4+16=20$ Or
$z=2\sqrt {5}$ mi
Taking derivative with respect to t of equation (1)
$\frac{d(z^2)}{dt}=\frac{d(x^2)}{dt}+\frac{d(y^2)}{dt}$
$\frac{d(z^2)}{dz}\frac{dz}{dt}=\frac{d(x^2)}{dx}\frac{dx}{dt}+\frac{d(y^2)}{dy} \frac{dy}{dt}$
$2z \frac{dz}{dt}=2x\frac{dx}{dt}+2y \frac{dy}{dt}$
$z \frac{dz}{dt}=x\frac{dx}{dt}+y \frac{dy}{dt}$ ........................... eq (2)
Putting $ x=2$ mi, $y=4$ mi, $z=2\sqrt {5}$ mi, $\frac{dx}{dt}=-600$mi/h, $\frac{dy}{dt}=-1200$ mi/h in equation (2)
$2\sqrt {5} \frac{dz}{dt}=-2 \times 600-4\times 1200=-6000 $ mi/h
$ \frac{dz}{dt}=-\frac{6000}{2\sqrt {5} }=-\frac{3000}{\sqrt {5}}=-600\sqrt 5\approx -1342$ mi/h
