Plane Flying ParametricallyDate: 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/ |
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