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A ship is sailing due south with a speed of 36 kilometers per hour.
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An iceberg lies 24 degrees north of east.
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After one hour, the ship is 33 degrees south of west of the iceberg.
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Find the distance between the ship and the iceberg at this time, giving the answer to the nearest kilometer.
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A lot of the skill involved in this question is in drawing the diagram.
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Let’s start with a compass showing the four directions.
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We’ll then take each statement separately and consider how to represent it.
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Firstly, we know that the ship is sailing due south.
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Initially, we’re told that an iceberg lies 24 degrees north of east of the ship’s starting point.
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Now, directly east would be directly to the right of the ship on our diagram.
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24 degrees north of east would mean the iceberg lies somewhere along the line like this.
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We’re then told that after one hour, the ship is 33 degrees south of west of the iceberg.
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Well, west would be the direction directly to the left of our iceberg.
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And using alternate angles in parallel lines, we know that the angle formed here is 24 degrees.
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So the full angle between the horizontal and the position the ship has now moved to is 33 degrees.
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And we can now see that we have a triangle.
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We can work out the angles in our triangle.
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For example, this angle here is the difference between 33 degrees and 24 degrees.
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It is nine degrees.
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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.
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The one piece of information we haven’t used yet is that the ship is traveling at a speed of 36 kilometers per hour.
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And we know it takes one hour for the ship to go from its original position to its new position.
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The ship will therefore have traveled 36 kilometers in this time.
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So we also know one side length in our triangle.
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What we were asked to calculate is the distance between the ship and the iceberg at this later time.
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So that’s this side here, which we can refer to as 𝑑 kilometers.
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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.
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Let’s look at the particular combination of information we’ve got.
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We know an angle of nine degrees and the opposite side of 36 kilometers.
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We also know an angle of 114 degrees.
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And we wish to calculate the opposite side of 𝑑 kilometers.
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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.
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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.
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𝑎 over sin 𝐴 equals 𝑏 over sin 𝐵, which is equal to 𝑐 over sin 𝐶.
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We only need to use two parts of this ratio.
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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.
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Our side 𝑑 is opposite the angle of 114 degrees, and the side of 36 kilometers is opposite the angle of nine degrees.
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So we have 𝑑 over sin of 114 degrees equals 36 over sin of nine degrees.
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We can solve this equation by multiplying each side by sin of 114 degrees, which is just a value.
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And it gives 𝑑 equals 36 sin 114 degrees over sin of nine degrees.
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Evaluating on a calculator, making sure our calculator is in degree mode, and we have 210.23267.
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The question asks us to give our answer to the nearest kilometer.
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So rounding appropriately, we have that the distance between the ship and the iceberg at this time is 210 kilometers.