Answer to Question #180470 in Calculus for ary

Question

1. An observer stands 800 meters from the launch pad of a rocket, watches it       ascend vertically at 100 m/s.

a.  Find the rate of change of the distance between the rocket and the observer. (Hint: Use Pythagorean Theorem)

b. Find the distance between the observer and the rocket when the rocket is 500 meters high.

c. What is the rate of change of the distance between the rocket and the observer when the rocket is 500 meters high?

Answer 1

let x be the vertical distance covered by the rocket from the launch pad and y be the distance between the rocket and obsever

$\therefore y^2=x^2+800^2(pythagorean) =y^2 =x^2+640000 ...(i)$

a) differentiating (i) implicitly with respect to time, we have

$2y \frac {dy}{dt}=2x \frac {dx}{dt} or \frac {dy}{dt}= \frac {2xdx}{2ydt}$

$=\frac xy (100)m/s = \frac{100x}{y}m/s$

$=\frac {100x}{\sqrt(x^2+640000)}m/s ...(ii)$

b) when x=500m, then;

$y^2=500^2+800^2\implies y= \sqrt {890000 }m= 100 \sqrt{89}m$

c)$\frac {dy}{dt}= \frac {100(500)}{100(\sqrt{89})}m/s = \frac {500}{\sqrt {89} } m/s$

$= 53m/s$ is the rate of change of the distance between the observer when the rocket is 500 m high from the launch pad