Video: Finding the Distance between Two Points

A ship is sailing due south with a speed of 36 km/h. An iceberg lies 24° north of east. After one hour, the ship is 33° south of west of the iceberg. Find the distance between the ship and the iceberg at this time, giving the answer to the nearest kilometer.

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Video Transcript

A ship is sailing due south with a speed of 36 kilometers per hour. An iceberg lies 24 degrees north of east. After one hour, the ship is 33 degrees south of west of the iceberg. Find the distance between the ship and the iceberg at this time, giving the answer to the nearest kilometer.

A lot of the skill involved in this question is in drawing the diagram. Let’s start with a compass showing the four directions. We’ll then take each statement separately and consider how to represent it. Firstly, we know that the ship is sailing due south. Initially, we’re told that an iceberg lies 24 degrees north of east of the ship’s starting point.

Now, directly east would be directly to the right of the ship on our diagram. 24 degrees north of east would mean the iceberg lies somewhere along the line like this. We’re then told that after one hour, the ship is 33 degrees south of west of the iceberg. Well, west would be the direction directly to the left of our iceberg. And using alternate angles in parallel lines, we know that the angle formed here is 24 degrees.

So the full angle between the horizontal and the position the ship has now moved to is 33 degrees. And we can now see that we have a triangle. We can work out the angles in our triangle. For example, this angle here is the difference between 33 degrees and 24 degrees. It is nine degrees. We could also work out this angle here, it’s 24 degrees, plus the angle between south and east, which is 90 degrees, giving a total of 114 degrees.

The one piece of information we haven’t used yet is that the ship is traveling at a speed of 36 kilometers per hour. And we know it takes one hour for the ship to go from its original position to its new position. The ship will therefore have traveled 36 kilometers in this time. So we also know one side length in our triangle.

What we were asked to calculate is the distance between the ship and the iceberg at this later time. So that’s this side here, which we can refer to as 𝑑 kilometers. We’ve now set up our diagram, and we see that we have a non-right-angled triangle, which means we’re going to apply either the law of sines or the law of cosines. Let’s look at the particular combination of information we’ve got.

We know an angle of nine degrees and the opposite side of 36 kilometers. We also know an angle of 114 degrees. And we wish to calculate the opposite side of 𝑑 kilometers. We therefore have opposite pairs of sides and angles, which tells us that we should be using the law of sines to answer this question.

Remember, this tells us that the ratio between each side length, represented using lowercase letters, and the sine of its opposite angle, represented using capital letters, is constant. 𝑎 over sin 𝐴 equals 𝑏 over sin 𝐵, which is equal to 𝑐 over sin 𝐶. We only need to use two parts of this ratio. And there’s no need to label our triangle using the letters 𝐴, 𝐵, and 𝐶 as long as we’re clear about what they represent.

Our side 𝑑 is opposite the angle of 114 degrees, and the side of 36 kilometers is opposite the angle of nine degrees. So we have 𝑑 over sin of 114 degrees equals 36 over sin of nine degrees. We can solve this equation by multiplying each side by sin of 114 degrees, which is just a value. And it gives 𝑑 equals 36 sin 114 degrees over sin of nine degrees. Evaluating on a calculator, making sure our calculator is in degree mode, and we have 210.23267.

The question asks us to give our answer to the nearest kilometer. So rounding appropriately, we have that the distance between the ship and the iceberg at this time is 210 kilometers.

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