A privately owned yacht leaves a dock in Myrtle Beach, South...

A privately owned yacht leaves a dock in Myrtle Beach, South Carolina, and heads toward Free port in the Bahamas at a bearing of S $1.4^{\circ} \mathrm{E}$ The yacht averages a speed of 20 knots over the 428 -nautical-mile trip. (a) How long will it take the yacht to make the trip? (b) How far east and south is the yacht after 12 hours? (c) A plane leaves Myrtle Beach to fly to Free port. At what bearing should it travel?

A privately owned yacht leaves a dock in Myrtle Beach, South Carolina, and heads toward Free port in the Bahamas at a bearing of S $1.4^{\circ} \mathrm{E}$ The yacht averages a speed of 20 knots over the 428 -nautical-mile trip.
(a) How long will it take the yacht to make the trip?
(b) How far east and south is the yacht after 12 hours?
(c) A plane leaves Myrtle Beach to fly to Free port. At what bearing should it travel?

Key Concepts

Bearing Interpretation and Conversion

Bearings provide a directional reference often measured in degrees from a cardinal direction. Converting a bearing such as 'S 1.4° E' into standard mathematical components is essential for accurately determining directional movement relative to north, south, east, or west.

Vector Decomposition

Vector decomposition involves breaking down a movement or force into horizontal and vertical components, which is particularly useful in navigation. By decomposing the travel distance into eastward and southward parts using the appropriate trigonometric functions, navigators can determine precise position changes over time.

Distance, Speed, and Time Relationship

This concept involves understanding the fundamental relationship between distance, speed, and time, often expressed in the formula time = distance ÷ speed. It is crucial for determining how long a journey takes when speed and distance are known.

Trigonometry in Navigation

Trigonometry is used in navigation to decompose movements into their component vectors. By applying trigonometric functions such as sine and cosine, one can calculate the eastward and southward displacements from a total distance traveled at a given bearing.

Transcript

00:01 Our bearing is s .1 .4e at 20 knots, and the total distance that this trip is is 428 nautical miles.

00:11 And so in part a, what we're going to figure out is how long it's going to take to actually travel this 428 nautical miles.

00:18 And so what we do is since 20 knots, that is equal to 20 nautical miles per hour.

00:24 We're just going to divide 428 nautical miles, which i'm just going to denote as nm, and we're going to divide it by 20 knots or times it by one hour divided by 20 knot, 20 nautical miles.

00:44 And so what this will do is it'll give us an answer in hours, which will be the amount of time that it actually takes to get to our.

00:54 End location or our destination.

00:57 So if we do this and we plug this into a calculator, 428 divided by 20 comes out to be 21 .4.

01:04 So it takes 21 .4 hours to actually do the whole trip.

01:11 So 21 .4 hours is the total amount of time it takes to actually sail from myrtle beach to the bahamas in this case.

01:22 And for part b, what we're going to figure out is how far, in the east direction and the south direction the yacht is after 12 hours so the first thing that we want to do is multiply 12 hours 20 nautical miles per hour or 20 knots and we do this we'll get an answer in nautical miles and this is going to be equal to 240 nautical miles and now that we have this total distance that our yacht is taken in this direction of our bearing.

02:00 What we can do is we can set up this bearing.

02:03 So we have this graph.

02:07 This is north.

02:08 This is south.

02:09 This is east and this is west.

02:12 And so our bearing, if we go south and then 1 .4 degrees to the east, it's going to be something like this, with this angle here being 1 .4 degrees.

02:26 And so we can, so we can, say that this distance, which is going to be the hypotenuse of our triangle, is 240 miles since we've gone 240 miles in this 12 hours.

02:38 And so what we can do now is we just have to figure out the length of the other two sides of our triangle.

02:45 So we want this length here and then this length down here as well, which we call b.

02:50 And so we're going to want to figure out one of the lengths and then the other one.

02:54 So the first one i'll do is the south or the amount of miles we traveled in the south direction.

03:00 And so we're going to want a trigonomic identity or trigonomic function, sorry, that actually relates adjacent sides as well as our hypotenuse, since we're given this angle here between the adjacent side and the hypotenuse.

03:13 So if we go back to sokatoa, cosine relates the adjacent side and the hypotenuse side...

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