Rate of change of distance from radar station to rocket

The radar station, launch pad, and rocket form a right triangle with legs $3000$ (horizontal, fixed) and $h$ (vertical, the rocket's height), and hypotenuse $x$ (distance from station to rocket). This gives the relating equation $x^2 = 3000^2 + h^2.$ Differentiating both sides with respect to $t$ using the chain rule gives $2x\frac{dx}{dt} = 2h\frac{dh}{dt},$ so $\frac{dx}{dt} = \frac{h}{x}\frac{dh}{dt}.$

At the instant in question, $h = 4000$ and $\frac{dh}{dt} = 600$. First find $x$ using the Pythagorean theorem: $x = \sqrt{3000^2 + 4000^2} = \sqrt{9{,}000{,}000 + 16{,}000{,}000} = \sqrt{25{,}000{,}000} = 5000$ ft. Substituting into $\frac{dx}{dt} = \frac{h}{x}\frac{dh}{dt}$ gives $\frac{dx}{dt} = \frac{4000}{5000}(600) = 480 \text{ ft/s}.$