DISTANCE A family is traveling due west on a road that...

DISTANCE A family is traveling due west on a road that passes a famous landmark. At a given time the bearing to the landmark is N $62^{\circ}$W, and after the family travels 5 miles farther the bearing is N $38^{\circ}$W. What is the closest the family will come to the landmark while on the road?

Key Concepts

Bearings and Compass Notation

Bearings are a method of expressing direction based on a reference line, usually true north. In compass notation, directions are given as angles measured from north or south toward east or west. This concept is crucial in navigation and aligns geometric problems with real-world directional movements.

Right Triangle Trigonometry

Right triangle trigonometry involves relationships between the angles and sides of a right-angled triangle using trigonometric ratios such as sine, cosine, and tangent. This concept is essential in solving navigation problems where angles and distances are used to determine positions and relationships between points.

Perpendicular Distance from a Point to a Line

The shortest distance from a point to a line is measured along the perpendicular from the point to the line. In the context of navigation and geographic positioning, determining this minimum distance often involves constructing and analyzing a right triangle within a geometric configuration, making it a fundamental concept for finding closest approach distances.

Transcript

00:01 The first thing in this problem that we want to do is draw a diagram to represent a situation.

00:04 So we're told that a family is traveling due west on a road that passes a famous landmark.

00:09 So let's say this is where the family starts and they're traveling due west.

00:14 So in this direction.

00:16 So they said at a given time the bearing to the landmark is nor 62 degrees west.

00:21 So let's say this is my landmark.

00:22 So that means if i drew in the distance between them and i put in my bearings here, so i put in my marks.

00:29 So what they said, it would be north 62 degrees west.

00:32 So that would be this angle right here.

00:35 So that's 62 degrees.

00:37 All right.

00:38 And then they said, after the family travels five miles further, meaning the distance between their initial point and their ending point is five miles, we're told that the bearing is north 38 degrees west.

00:50 So the first thing i'm going to do is i'm going to make this shorter because if it's still northwest, that means it hadn't gone that far.

00:58 So let's say we end up here.

01:00 But again, this is still five miles.

01:03 So let's put in our line here.

01:06 So now we're told that this bearing is north 38 degrees west.

01:10 So meaning if i have this here, that means that this angle right here, the small part, is 38 degrees.

01:19 All right.

01:19 We want to find what is the closest the family will come to the landmark while on the road.

01:25 Okay.

01:26 So in order to do this, what we need to do is we need to find the distance that the landmark would be from our road.

01:35 So essentially we're trying to find this distance off to the side.

01:38 Okay.

01:39 Well, in order to do that, we might have to find a couple of pieces of information first.

01:43 Well, first off, we know that this angle is 62 degrees.

01:48 This angle it forms is 90 degrees.

01:51 So we can find the smaller angle by subtracting 90 minus 62, which is 28.

01:56 So we know that this angle and our triangle is 28 degrees.

02:01 Okay.

02:03 So now let's think about the other angle on this side.

02:07 Well, we know that this whole angle is going to be 90 degrees...

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