Expert: Alon Mandes - 5/16/2009

Question
An airplane flying horizontally at an altitude of 3 miles and at a speed of 480mi/hr passes directly above an observer on the ground. How fast is the distance from the observer to the airplane increasing 30 seconds later?

this is my first of R-R so can you explain in detail and so far i got:
If the plane is moving at 480 mi/hr, use 30 seconds to compute how far it traveled.  (480 miles/hour)(hour/60 minutes) = 8 miles/minute.
It can be seen that 30 seconds is half a minutes, so the plane would have flown half of 8 miles.  You can probably do this, but 8/2=4.
If the plane were 3 miles overhead and travelled 4 miles away, the distance can be found as √(3²+4²).  Compute 3².  Compute 4².  Add the two together.  The number should be recognized as X².
and rest is kind of confusing so plese help


Answer
Your direction is correct but need to be careful :
According to Pythagoras x=√[a(t)²+b(t)²]. Where a(t) is the vertical distance, & b(t) is the horizontal distance. It's easy to see that
a(t)=3 miles & b(t)=8t miles . Note that b is function of time !
Thus : x(t)=√[3²+(8t)²]=√[9+64t²]. To find How fast the distance is
increasing we calculate the derivative of x(t) , this will give us
the rate of change :
x'(t)=64t/√[9+64t²]
x'(t=½)=64*½/√[9+64*¼]=32/√[9+16]=32/5 .

Alon.