Related Rates Problems

Date: 01/15/99 at 23:11:48
From: Hiu Sze
Subject: Related Rate

1) A plane flying horizontally at an altitude of 1 mi and a speed of 
500 mi/h passes directly over a radar station. Find the rate at which 
the distance from the plane to the station is increasing when it is 2 
miles away from the station.

Answer: The plane's altitude is 1 mile and its velocity is 500 mi/h.

2) At noon, ship A is 150km west of ship B. Ship A is sailing east at 
35 km/h and ship B is sailing north at 25 km/h. How fast is the 
distance between the ships changing at 4:00 p.m.?

Thank you!
Hiu Sze

Date: 01/16/99 at 10:31:31
From: Doctor Anthony
Subject: Re: Related Rate

Question 1:

Let x = the horizontal distance that the plane travels from a point 
directly overhead. Let y = the slant distance from the radar station to 
the plane.

Then by the Pythagorean Theorem, y^2 = x^2 + 1^2 and differentiating 
with respect to t:

   2y(dy/dt) = 2x(dx/dt) and dx/dt = 500

   dy/dt = (x/y)(500)

When x = 2, y = sqrt(4+1) = sqrt(5), so  x/y = 2/sqrt(5) and

   dy/dt = 1000/sqrt(5) = 447.21 mph

So the distance between the plane and the radar station is increasing 
at 447.21 mph.
 
Question 2:

Let x = the distance of ship A west of the north-south line through B.
    y = the distance of ship B north of the east-west line through A.

At any time the distance apart is given by L^2 = x^2 + y^2.

Then:

   2L dL/dt = 2x dx/dt + 2y dy/dt
    L dL/dt = x(-35) + y(25)
      dL/dt = [-35x + 25y]/L

At 4:00 p.m.:

          x = 150-140 = 10     
          y = 100     
        L^2 = 100+10000 = 10100   =>   L = 100.4988

So:

      dL/dt = [-350 + 2500]/100.4988 = 21.393 km/hr

So at 4:00 p.m., the ships are separating at a speed of 21.393 km/hr.

- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/