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A man standing on the deck of a ship, 10 meters above water level, observes the angle of elevation of the top of a hill as 60 degrees and angle of depression of the base of the hill as 30 degrees.
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Find the distance of the hill from the ship and the height of the hill.
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This is a geometry problem.
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And so our first step is to draw a diagram to help us see what’s going on.
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We’re told our observer is 10 meters above water level.
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So let’s draw in water level and put our man 10 meters above this level.
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The fact that he’s standing on the deck of a ship isn’t important to the geometry of the situation.
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So we avoid drawing the ship.
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He observes the angle of elevation of the top of the hill as 60 degrees.
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So we draw in the horizontal from where the man is standing.
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The angle of elevation is 60 degrees.
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So the top of the hill lies on the line inclined at 60 degrees to the horizontal.
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Let’s call it 𝑇 and put it here.
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And the angle of depression of the base of the hill is 30 degrees.
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So this is 30 degrees below the horizontal.
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And of course, the base of the hill, which we’ll call 𝐵, is at water level.
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Notice that I’ve carefully drawn the diagram so that the top of the hill lies vertically above the bottom of the hill.
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We’re not explicitly told this in the question.
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But if we don’t make this assumption, then we don’t have enough information to solve the question.
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We’re given a hint that we have to make this assumption in the question though because we’re asked for the distance of the hill from the ship.
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And if we make the assumption that the hill is vertical, then it’s clear that this distance is a horizontal distance, 𝑀𝑃.
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But if we don’t make this assumption, then it’s more difficult to work out which distance we have to measure.
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In addition to this distance which I’ve called 𝑑, we also have to find the height of the hill which I’ll call ℎ.
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We need to explain the notation that we’ve used.
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𝑀 is the point from which the man is observing.
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𝑇 is the top of the hill and 𝐵 is its bottom.
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And 𝑃 is a point on the hill which we’ve decided is vertical, which is at the same height as the man.
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We first work out the distance 𝑑 of the hill from the ship which is of course the same as the distance of the hill from the man who’s on the ship.
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If we consider triangle 𝑀𝑃𝐵, which is right angled as the vertical hill 𝑇𝐵 is perpendicular to the horizontal 𝑀𝑃, we can see that the side 𝑃𝐵 is 10 meters long by definition.
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And now, we can apply some trigonometry.
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Tan of 30 degrees is the length of the side opposite the angle of 30 degrees divided by the length of the side adjacent to the angle of 30 degrees.
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In triangle 𝑀𝑃𝐵, the opposite side is side 𝑃𝐵 and the adjacent side is side 𝑀𝑃.
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We know that 𝑃𝐵 is 10 meters.
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And the length 𝑀𝑃 is the distance 𝑑 we’re looking for.
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But we also know the value of the left-hand side, tan 30 degrees, because 30 degrees is a special angle.
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Tan 30 degrees is one over square root three.
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And now, we can rearrange this to find our distance 𝑑.
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Multiplying both sides by 𝑑, we get that 𝑑 over root three is 10.
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And multiplying both sides by root three, we find that our distance 𝑑 is 10 times root three.
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And of course, this distance is measured in meters.
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The distance is 10 root three meters.
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So let’s add this value to our diagram and clear some room to work on the next part of the question, which is to find the height ℎ of the hill.
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We can see from the diagram that the distance ℎ, being the length of the line segment 𝑇𝐵, is 𝑇𝑃 plus 𝑃𝐵.
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And we already know the value of 𝑃𝐵.
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It is 10 meters.
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And so the height ℎ is 𝑇𝑃 plus 10.
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Our task, therefore, is to find the length of 𝑇𝑃.
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And look, we have another right-angled triangle, the triangle 𝑇𝑃𝑀.
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We know the value of an angle in this right-angled triangle, 60 degrees.
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And we know the length of the side adjacent to this angle, that’s 𝑀𝑃 with a length of 10 root three.
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We’re looking for the length of the side opposite this angle.
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So again, we use a tangent ratio, tan of 60 degrees, is the length of the opposite side 𝑇𝑃 over the length of the adjacent side 𝑀𝑃.
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60 degrees is a special angle.
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And so we know tan 60 degrees, it’s root three.
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And we also know that 𝑀𝑃 is 10 root three.
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We found that earlier.
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What we’re looking for is 𝑇𝑃.
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And we can find this by rearranging our equation.
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We multiply both sides by 10 root three.
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And as root three times root three is three, we find that 𝑇𝑃 is 10 times three which is 30.
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The reason we were looking for 𝑇𝑃 is because we knew that the height we’re looking for was 10 more than 𝑇𝑃.
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We see, therefore, that the height ℎ is 30 plus 10 which is 40.
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And again, this is measured in meters.
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We should write down this conclusion explicitly, interpreting the value of ℎ in the context of the question.
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To conclude, we found that the distance of the hill from the ship is 10 times root three meters.
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And the height of the hill is 40 meters.
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Once we’d drawn the correct diagram with the assumption that the top of the hill lies vertically above the bottom, this question was simply a matter of applying trigonometry to right-angled triangles with special angles.