Location of a fire two fire towers are 30 kilometers apart...

LOCATION OF A FIRE Two fire towers are 30 kilometers apart, where tower $A$ is due west of tower $B$. A fire is spotted from the towers, and the bearings from $A$ and $B$ are N$76^\circ$E and N$56^\circ$W, respectively (see figure). Find the distance $d$ of the fire from the line segment $AB$.

Transcript

00:04 All right, so in this particular problem, we have two fire towers, and we know that the fire towers are 30 kilometers apart.

00:17 We have fire tower a and b.

00:23 We know that a is due west of the b, so that we know a is on the left.

00:28 And then we have a fire that is spotted from the towers, and the bearings from a and b are north.

00:35 76.

00:38 So if we go north, 76 degrees east, that is the location of the first fire.

00:46 So north from a and then 76 degrees to the east.

00:53 So here.

00:56 And then from tower b is north 56 degrees west.

01:01 So we're going to go north and then 56 degrees to the west and that's where the fire is.

01:18 And then the question asks, find the distance d of the fire from the line a, b.

01:25 So we're looking for this distance here.

01:29 All right, so what do we know? well, we know that if this is a 76 degree angle then we know that this is a 14 degree angle and i can already tell that i'm not this is not drawn much to scale but that's okay this is a 56 degree angle then this is going to be a 34 degree angle so we've got that and then we're going to use the law of signs to help us solve this so if i can find either it doesn't really matter which one, but i'm going to call this side b and i'll call this side a since this is opposite a.

02:13 So if i can find either a or b, then i can use right triangle trig to find that missing d side.

02:20 So it doesn't really matter which side i want to which side to use.

02:24 But if this is 14 degrees and this is 34 degrees, then i know that this angle must be 132 degrees.

02:37 So that it's the whole 180 degrees of the triangle.

02:41 So then i'm going to use the law of signs on that large triangle.

02:46 So i'm going to say the sign of 14 degrees over a is equal to the sign of 132 degrees over 30...

Question

LOCATION OF A FIRE Two fire towers are 30 kilometers apart, where tower $A$ is due west of tower $B$. A fire is spotted from the towers, and the bearings from $A$ and $B$ are N$76^\circ$E and N$56^\circ$W, respectively (see figure). Find the distance $d$ of the fire from the line segment $AB$.

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