NAVIGATION A privately owned yacht leaves a dock in Myrtle...

NAVIGATION A privately owned yacht leaves a dock in Myrtle Beach, South Carolina and heads toward Freeport in the Bahamas at a bearing of S $1.4^\circ$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) If a plane leaves Myrtle Beach to fly to Freeport, what bearing should be taken?

NAVIGATION A privately owned yacht leaves a dock in Myrtle Beach, South Carolina and heads toward Freeport in the Bahamas at a bearing of S $1.4^\circ$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) If a plane leaves Myrtle Beach to fly to Freeport, what bearing should be taken?

Key Concepts

Inverse Trigonometry for Bearing Calculation

When the components of displacement are known, inverse trigonometric functions can be applied to determine the overall angle or bearing of the travel. For instance, given eastward and southward distances, one can use the arctangent function to determine the angle that the resultant displacement makes with one of the cardinal directions, thereby converting rectangular coordinates back into a navigational bearing.

Vector Decomposition Using Trigonometry

When a movement is described by a bearing and a distance, it can be useful to break the movement into its orthogonal components, typically east and south (or north and east, etc.). This is done using trigonometric functions: the sine function gives one component while the cosine function gives the other based on the angle provided by the bearing. This decomposition allows for the separate analysis of displacement in different directions, which is essential in many navigation and physics problems.

Speed?Distance?Time Relationship

This concept relates the three fundamental quantities in motion: speed (the rate of travel), distance (the length of the journey), and time (the duration of travel). In navigation and other contexts, if any two of these quantities are known, the third can be calculated using the formula time = distance/speed. This relationship is particularly useful for estimating travel durations in contexts such as marine navigation where speeds are often given in knots and distances in nautical miles.

Bearings and Navigational Angles

Bearings are used in navigation to express direction relative to a fixed reference such as north or south. They often involve a combination of a cardinal direction with an angular deviation, such as 'S 1.4° E', which means to start from due south and deviate by 1.4° toward the east. Understanding this notation is crucial for plotting courses and for aligning navigational instruments to ensure the intended direction is followed.

Transcript

00:03 Okay, the given information in this problem is that the yacht is traveling at 20 knots, which means 20 nautical miles per hour.

00:12 We know the trip it makes is 428 nautical miles, and we know the bearing is south 1 .4 degrees east.

00:20 And in part a, we're finding how long it takes the yacht to make the trip.

00:24 So we know the trip is 428 nautical miles, and we're going to multiply that by one hour per 20 nautical.

00:34 Miles based on our rate of 20 knots.

00:38 So essentially we're dividing our distance by 20, and that gives us 21 .4 hours.

00:47 If you want to put that in hours and minutes, that would be 21 hours and 24 minutes for this trip.

00:56 Okay, let's move on to part b.

01:00 Go ahead and circle that answer.

01:01 Okay, in part b, we're going to figure out how far east and south the yacht is after 12 hours.

01:12 So if it's traveling for 12 hours at 20 nautical miles per hour, then it goes a total of 250 nautical miles.

01:24 Excuse me, 240 nautical miles.

01:26 Okay, so let's draw a picture of the situation.

01:30 So let's draw our little compass, and then we know we need a 1 .4 degree angle, so obviously not to scale, but we'll just make it kind of like that.

01:40 So this angle here is 1 .4 degrees east of south, and the distance traveled is 240.

01:48 And so we're figuring out how far south and east this point down here is from the original point.

01:53 So let's complete a right triangle.

01:56 And we can label the horizontal distance x and the vertical distance y, and we're solving for x and y.

02:02 So we know that if this angle here is 1 .4 degrees, then so is this angle here.

02:07 We have alternate interior angles with parallel lines cut by a transversal.

02:13 So we can set up some equations such as the sign of 1 .4 degrees is equal to opposite over hypotenuse, x over 240, and then we can go ahead and multiply both sides of that equation by 240...

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