### Video Transcript

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. Find the distance of the hill from
the ship and the height of the hill.

This is a geometry problem. And so our first step is to draw a
diagram to help us see what’s going on. We’re told our observer is 10
meters above water level. So let’s draw in water level and
put our man 10 meters above this level. The fact that he’s standing on the
deck of a ship isn’t important to the geometry of the situation. So we avoid drawing the ship.

He observes the angle of elevation
of the top of the hill as 60 degrees. So we draw in the horizontal from
where the man is standing. The angle of elevation is 60
degrees. So the top of the hill lies on the
line inclined at 60 degrees to the horizontal. Let’s call it 𝑇 and put it
here. And the angle of depression of the
base of the hill is 30 degrees. So this is 30 degrees below the
horizontal. And of course, the base of the
hill, which we’ll call 𝐵, is at water level.

Notice that I’ve carefully drawn
the diagram so that the top of the hill lies vertically above the bottom of the
hill. We’re not explicitly told this in
the question. But if we don’t make this
assumption, then we don’t have enough information to solve the question. 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.

And if we make the assumption that
the hill is vertical, then it’s clear that this distance is a horizontal distance,
𝑀𝑃. But if we don’t make this
assumption, then it’s more difficult to work out which distance we have to
measure. 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.

We need to explain the notation
that we’ve used. 𝑀 is the point from which the man
is observing. 𝑇 is the top of the hill and 𝐵 is
its bottom. And 𝑃 is a point on the hill which
we’ve decided is vertical, which is at the same height as the man.

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. 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. And now, we can apply some
trigonometry.

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. In triangle 𝑀𝑃𝐵, the opposite
side is side 𝑃𝐵 and the adjacent side is side 𝑀𝑃. We know that 𝑃𝐵 is 10 meters. And the length 𝑀𝑃 is the distance
𝑑 we’re looking for. But we also know the value of the
left-hand side, tan 30 degrees, because 30 degrees is a special angle. Tan 30 degrees is one over square
root three.

And now, we can rearrange this to
find our distance 𝑑. Multiplying both sides by 𝑑, we
get that 𝑑 over root three is 10. And multiplying both sides by root
three, we find that our distance 𝑑 is 10 times root three. And of course, this distance is
measured in meters. The distance is 10 root three
meters. 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.

We can see from the diagram that
the distance ℎ, being the length of the line segment 𝑇𝐵, is 𝑇𝑃 plus 𝑃𝐵. And we already know the value of
𝑃𝐵. It is 10 meters. And so the height ℎ is 𝑇𝑃 plus
10. Our task, therefore, is to find the
length of 𝑇𝑃. And look, we have another
right-angled triangle, the triangle 𝑇𝑃𝑀. We know the value of an angle in
this right-angled triangle, 60 degrees. And we know the length of the side
adjacent to this angle, that’s 𝑀𝑃 with a length of 10 root three. We’re looking for the length of the
side opposite this angle.

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. 60 degrees is a special angle. And so we know tan 60 degrees, it’s
root three. And we also know that 𝑀𝑃 is 10
root three. We found that earlier. What we’re looking for is 𝑇𝑃. And we can find this by rearranging
our equation. We multiply both sides by 10 root
three. And as root three times root three
is three, we find that 𝑇𝑃 is 10 times three which is 30.

The reason we were looking for 𝑇𝑃
is because we knew that the height we’re looking for was 10 more than 𝑇𝑃. We see, therefore, that the height
ℎ is 30 plus 10 which is 40. And again, this is measured in
meters. We should write down this
conclusion explicitly, interpreting the value of ℎ in the context of the
question.

To conclude, we found that the
distance of the hill from the ship is 10 times root three meters. And the height of the hill is 40
meters. 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.