A ferry transports tourists between three islands. It sails...

A ferry transports tourists between three islands. It sails from the first island to the second island, $4.76 \mathrm{~km}$ away, in a direction $37.0^{\circ}$ north of east. It then sails from the second island to the third island in a direction $69.0^{\circ}$ west of north. Finally it returns to the first island, sailing in a direction $28.0^{\circ}$ east of south. Calculate the distance between (a) the second and third islands and (b) the first and third islands.

Key Concepts

Magnitude Calculation using the Pythagorean Theorem

After obtaining the resultant vector from the addition of its components, its magnitude, or the distance, is calculated using the Pythagorean theorem. This involves squaring the horizontal and vertical components, summing them, and then taking the square root of the result. This method is fundamental in determining the direct distance between two points in a Cartesian plane.

Vector Addition

Once vectors are decomposed into their components, vector addition is performed by summing the corresponding components along each axis. This operation yields the resultant vector, which represents the overall effect of multiple displacements or forces. In problems involving multiple legs of a journey, this concept is essential for determining the net movement or displacement between points.

Vector Decomposition into Components

This concept involves breaking a vector into its horizontal and vertical components, which simplifies the analysis of movements in two dimensions. In navigation and physics problems, vectors such as displacements are often described in terms of magnitude and a direction (angle). Decomposing a vector into its components using trigonometric functions (sine and cosine) allows one to add vectors algebraically, making it easier to calculate resultant displacements or distances.

Navigational Angles and Bearing Conversions

Navigational problems typically describe directions using cardinal references (north, south, east, west) and relative angles (e.g., north of east, west of north). Understanding how to convert these bearings into standard angles or component form is crucial. This conversion enables one to apply trigonometry correctly by aligning the directions with a coordinate system, usually taking east as the positive x-axis and north as the positive y-axis.

Transcript

00:01 In this question, we have three islands, one, two, and three.

00:04 And then there's a ferry that goes from island 1 to 2, island 2, and then island 2 to island 3.

00:12 And then from island tree back to island 1.

00:15 Okay.

00:16 So we are given that.

00:18 So this is island 1, and then it goes to island 2.

00:25 Okay.

00:27 And in a direction that is 37 degrees north of east okay and the distance is 4 .76 kilometers then the ferry goes from 2 to 3 the direction is 69 degrees west of north so it's like this from north to west okay 59 degrees from island three back to one the direction is 28 degrees east of south okay so this is how the setup would look like and then first thing is to identify the unknown all the angles okay so yeah because in this question we want to find the distance between island two and three and island one other distance between island one and island three okay so to do that we need to first identify the angles in the triangle okay so draw this vertical line here we can know that this is 28 degrees and this is 53 degrees right and then we draw another horizontal line here we can know we are able to tell that this is 37 degrees and then here is 21 degrees okay and so when you add up all the angles at the bottom two side and then you are two corners and then you can tell that this angle here is 47 degrees okay so to find the distance between in part a you want to find the distance between island two and island three of the second and third island so we can use the law of science use the law of signs okay so can sum up the angle here so this is 81 degrees this is 58 degrees okay so 81 48 8 8 the angle here is not 47 the angle here should be 1 3 9 so should be 41 degrees okay okay alright, so using a lot of signs, we have sign 41 degrees divided by d12.

03:17 It goes to sine 81 degrees by d23...

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