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Plane Flying Parametrically


Date: 06/28/99 at 15:58:13
From: Jason 
Subject: Plane flying parametrically

I have been trying to figure out how to get the closest result. I 
don't know which answer would be correct. The problem is:

The position of an airplane approaching its airport is described 
parametrically by P_t = (1000,500,900)+ t[-100,-50,-90]. For what 
value of t is the airplane closest to the traffic control center 
located at (24,11,13)?


Date: 06/29/99 at 08:28:39
From: Doctor Jerry
Subject: Re: Plane flying parametrically

Hi Jason,

Thanks for your question.

The x-, y-, z-coordinates of the plane at time t are

     x = 1000-100t
     y = 500-50t
     z = 900-90t.

Often, they are put into vector form, so that the position vector of 
the plane at time t is

     {x,y,z} = {1000-100t,500-50t,900-90t}

The position vector of the traffic control center is {24,11,13}. You 
want the distance from the line to this point. This can be done by 
minimizing a function using calculus.

By calculus, minimize

     D(t)=((1000-10t)-24)^2+((500-50t)-11)^2+((900-90t)-13)^2.

If you set D'(t)=0 and solve for t you will find t=10.6579, approx.

- Doctor Jerry, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Calculus
High School Physics/Chemistry

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