Solve each problem. Two boats leave a dock together. Each...

Solve each problem.
Two boats leave a dock together. Each travels in a straight line. The angle between their courses measures $54^{\circ} 10^{\prime} .$ One boat travels 36.2 kilometers per hour and the other 45.6 kilometers per hour. How far apart will they be after 3 hours?

Key Concepts

Uniform Linear Motion

This concept refers to motion at a constant speed, where the distance traveled is found by multiplying the constant speed by the time elapsed. It is essential in problems where objects move in a straight line without acceleration, allowing a straightforward calculation of the distance each object covers.

Law of Cosines

The Law of Cosines is an important rule in trigonometry used to determine the unknown side of a triangle when two sides and the included angle are known. It generalizes the Pythagorean theorem to any triangle, making it a valuable tool for solving problems involving non-right triangles formed by paths or vectors.

Angle Measurement and Conversion

Problems often present angles in various formats, such as degrees and minutes. Converting these measurements into a standard format, typically decimal degrees, is crucial for ensuring accuracy when applying trigonometric formulas like the Law of Cosines.

Transcript

00:01 So the best way to start this problem is to draw a picture.

00:03 So we're told that two boats are going to leave a dock at the same time.

00:07 So let's say that this right here is going to be our dock.

00:09 I'll call it empty.

00:10 Now, we're told that each is going to travel in a straight line, and we know that the angle between their courses measures 54 degrees and 10 minutes.

00:18 So let's say that one boat is going in this direction, and the other is heading in this direction.

00:22 We know that this angle right here is 54 degrees and 10 minutes.

00:28 And let me make those symbols a little bit clearer there.

00:31 So 54 degrees, 10 minutes.

00:34 Now, we're told that one boat is going to travel 36 .2 km per hour, and the other is going 45 .6 km per hour.

00:41 We want to figure out how far apart they are after three hours.

00:45 Well, because we know how fast they're going and how long they're going for, we can find the distance.

00:50 So let's say that this section is that first boat that's going 36 .2 km per hour.

00:56 So if i multiply that by 3, that will tell us our distance.

00:59 Well, 36 .2 times 3 is equal to 108 .6.

01:03 So that's how many kilometers the first boat has gone.

01:07 Now, the second boat is going 45 .6 km per hour.

01:10 So if we take 45 .6 and multiply by 3, that will tell us the distance the second boat goes.

01:16 Well, 45 .6 times 3 is equal to 136 .8...

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