A plane from Denver, Colorado, (altitude 1650 meters) flies...

A plane from Denver, Colorado, (altitude 1650 meters) flies to Bismark, North Dakota (altitude 550 meters). It travels at $650 \mathrm{km} /$ hour at a constant height of 8000 meters above the line joining Denver and Bismark. Bismark is about $850 \mathrm{km}$ in the direction $60^{\circ}$ north of east from Denver. Find parametric equations describing the plane's motion. Assume the origin is at sea level beneath Denver, that the $x$ -axis points east and the $y$ -axis points north, and that the earth is flat. Measure distances in kilometers and time in hours.

Transcript

00:01 So in this question we're told that we start out at denver, which has an x and y position of zero.

00:10 So we're saying that it's zero in the x and y coordinates, but it has an altitude of 1 .65 kilometers.

00:22 And we then have bismarck, which is at an altitude of, it's at an altitude of 550 metres, so 0 .55 kilometers.

00:44 But then if we have denver here and bismarck here, then it's at a angle of 60 degrees north of east, and it's 850 kilometers.

00:58 So its x coordinate is going to be 850 times the cosine of 60 degrees, which is 425 kilometers.

01:16 And its y coordinate is going to be 850 times the sign of 60 degrees, which is 700 and 736 .12 kilometers.

01:34 So the plane travels at 650 kilometres per hour at a constant height of 8 ,000 meters above that line.

01:55 So that means that it starts off at an initial position of it has the x and y coordinates of denver, but it has the z coordinate, which is the height of denver plus 8 ,000 meters.

02:12 So that's going to be 9 .65 kilometers.

02:17 And then its final position is going to be the position of bismarck 425 kilometers, 736 .12 kilometers, and at a height of 0 .55 plus 8, which is 8 .55 kilometers.

02:45 And we want to get parametric equations describing the plane's motion.

02:50 So remember as well that v, the magnitude, its velocity is 650 kilometers per hour.

03:01 So that means that it's going to be traveling in the direction.

03:12 So x of t is going to be x0 plus x1 minus x0 divided by the magnitude of x1 minus x0 times times v because the times this magnitude v because we don't know what direction well we know what direction the velocity is in it's in the direction connecting these two things so i make a unit vector out of that direction and then multiply it by the magnitude of the velocity so that is going to give me parametric equations so i'm going to do this as a column vector because it's going to be easier so we've got zero here so it's just going to be x1 minus x -0.

04:05 So first of all, let's work out, let's work out what this is.

04:08 So this is going to be x -naut...

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