### Video Transcript

A surveyor walks through a field as
shown in the diagram. How much further east does the
surveyor walk than he walks north? Round your answer to the nearest
meter.

Okay, so we’ve been told in this
question that a surveyor is walking through a field. And the path that the surveyor
takes is shown in the diagram. So the surveyor starts here at the
origin of the axis that we’ve drawn and finishes here. The total distance that he walks is
450 meters as we’ve been told in the diagram. We also know that he walks at an
angle of 30 degrees to east, where east is this way — once again shown to us in the
diagram.

What we’ve been asked to do is to
find out how much further east does the surveyor walk than he walks north. So what does this mean? Well, we know that he walks in this
direction. Now, that direction can be broken
up into an eastwards component and a northwards component. Specifically, the eastwards
component is this distance here because that’s how much further east this point is
relative to this point. And similarly, the northwards
component is this distance here because you guessed it that’s how much further north
this point is relative to this point.

Now, what the question wants us to
do is to work out how much further east the surveyor walks compared to how far he
walks north. In other words, how much further is
this distance compared with this distance, which means we know we need to find these
distances now. So let’s give each one a name. Let’s call this one 𝑥 and this one
𝑦. So how far he walks east is 𝑥 and
how far he walks north is 𝑦.

Now, an important thing to know
about compass directions east and north is that they’re at right angles to each
other. Therefore, this is a right
angle. And so we’ve got ourselves a
right-angled triangle. The distance that he walks — the
450 meters — and the two components — the eastwards and the northwards components —
form the other two sides. So we can draw this triangle a lot
more simply. So here’s our right-angled triangle
with this being the right angle. Now, we want to work out the
distances 𝑥 and 𝑦.

We know one angle and the length of
the hypotenuse. So we need to use what we’ve learnt
in maths. We need to use SOHCAHTOA. Let’s first try to find the value
of 𝑥 using SOHCAHTOA. Well, the angle that we’ll be
working with is this one here. Now, relative to that angle, 𝑥 is
the adjacent side. And we already know the hypotenuse,
which is 450 meters. So in this case, we’re trying to
work out the adjacent and we know the hypotenuse. So we need to use cosine.

In other words, we need to use CAH
out of SOHCAHTOA. The CAH part tells us that the
cosine of an angle 𝜃 is equal to the adjacent 𝐴 divided by the hypotenuse 𝐻. Now, in this case, we already know
what 𝜃 is. We know that 𝜃 is 30 degrees. That’s the angle we’re working
with. And as well as this, we want to
work out what the adjacent is, which we know is 𝑥. So we replace the 𝐴 for adjacent
with 𝑥 and we also know the hypotenuse which is 450 meters.

What this means is that we can
calculate the left-hand side by plugging in to our calculator. And we know the hypotenuse. So we can work out what 𝑥 is. To do this though, we need to
rearrange the equation. What we can do is to multiply both
sides of the equation by 450, meaning the 450 is on the right-hand side cancel. And so what we’re left with is that
450 times cosine of 30 degrees is equal to 𝑥. We can then plug this in to our
calculator to give us 𝑥 is equal to 389.7114 dot dot dot meters.

So let’s write that down on the
bottom right-hand side of the screen and let’s now try and work out what the value
of 𝑦 is. Now relative to the angle 30
degrees, 𝑦 is the opposite side. And once again, we already know the
hypotenuse. In other words, we’re trying to
calculate what the opposite is and we know the hypotenuse. So we need to use sine.

We need to use SOH from
SOHCAHTOA. SOH tells us that the sine of the
angle 𝜃 — once again this 𝜃 is going to be 30 degrees — is equal to the opposite
length divided by the hypotenuse. And so we can do a similar thing to
what we did earlier. Sine of 30 degrees where 30 degrees
is 𝜃 becomes the opposite which is 𝑦 divided by the hypotenuse which is 450
meters. And yet again, we can rearrange by
multiplying both sides of the equation by 450 so that 450 cancels on the right-hand
side, leaving us with 450 times sine of 30 degrees is equal to 𝑦. We can plug this into our
calculator to give us 𝑦 is equal to 225 meters. So we write that down on the bottom
right.

Now what we want to do in this
question is to work out how much further east the surveyor walks compared to how far
he walks north. In other words, how much larger is
𝑥 compared to 𝑦? And the way to do this is to work
out what the value of 𝑥 minus 𝑦 is because 𝑥 minus 𝑦 is how much larger 𝑥 is
relative to 𝑦. Now happily, we already know what
𝑥 is and we know what 𝑦 is. So we can plug in the values. 389.7114 dot dot dot is 𝑥 and 225 is 𝑦 and
this evaluates to 164.7114 dot dot dot meters.

So is that our final answer? Well, no, we’ve been told to round
our answer to the nearest meter. In order to do this, we need to
round the last value before the decimal point. That’s the four. Now, it’s the number after that
that will tell us whether the value rounds up or stays the same. So in this case, we’ve got a seven
and seven is larger than five. So this value will round up. This will now become a five.

And this leads us to our final
answer. 𝑥 minus 𝑦 — in other words the
amount of distance the surveyor walks further east compared to how far he walks
north — is equal to 165 meters to the nearest meter.