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.
