Two ships are at sea on either side
of a lighthouse, such that the ships and the lighthouse are all lying on the same
straight line. The angles of depression of the two
ships, as observed from the top of the lighthouse, are 60 degrees and 45
degrees. If the height of the lighthouse is
200 metres, find the distance between the two ships. Take root three is equal to
Let’s begin by drawing a sketch of
the ships and lighthouse. Remember, a sketch does not need to
be to scale, but it should be roughly in proportion so that we can use it to figure
out how best to answer the question. Remember, the angle of depression
of a ship from the lighthouse is the angle between the horizontal and the line of
sight of the ship from the top of the lighthouse.
Since the sea is parallel to the
horizontal, we can include the angle between the line of sight and the sea. Alternate angles are equal, so it’s
60 degrees. Similarly, the angle between the
line of sight of the top of the lighthouse from the second ship and the sea is 45
We can assume that the lighthouse
is perpendicular to the sea. And in doing so, we can see that we
have two right-angled triangles, for which we know the measure of an additional
angle and the length of one of the sides. We can use right angle trigonometry
to calculate missing lengths in these triangles.
Let’s consider the triangle made by
the first ship and the lighthouse. Labelling the triangle, we can see
that the side representing the line of sight is the hypotenuse. That’s the longest side of the
triangle, and it can be found by looking for the side directly opposite the right
The opposite side is the side
representing the lighthouse; it’s the side opposite the given angle. The adjacent side is the other
side. This time, it’s the one next to the
included angle, and it’s the distance between the lighthouse and the first ship.
Since we know the length of the
opposite side and we’re looking to calculate the length of the adjacent, we use the
tangent ratio. Tan of 𝜃 is equal to opposite over
adjacent. Substituting what we know into this
formula, and we get tan of 60 is equal to 200 divided by 𝑥 one. Tan of 60 is equal to root three,
so our equation becomes root three is equal to 200 over 𝑥 one. And to solve, we first multiply
both sides by 𝑥 one. That gives us root three 𝑥 one is
equal to 200. Next, we divide through by root
three, and we get that 𝑥 one is equal to 200 over root three.
Now, at this point, it’s sensible
to rationalise the denominator. And we do so by multiplying by the
numerator and the denominator of this fraction by root three. This just creates an equivalent
fraction. But in doing so, we end up with
just three on the denominator, which is a rational number. We get 𝑥 one is equal to 200 root
three over three. The distance between the first ship
and the lighthouse is 200 root three over three metres.
We’re now going to repeat this
process to find the distance between the second ship and the lighthouse. Once again, we know the length of
the opposite side. And we’re looking to calculate the
length of the adjacent, so we’ll use the tangent ratio. This time, we get tan of 45 is
equal to 200 over 𝑥 two. Tan of 45 is equal to one, so we
get one is equal to 200 over 𝑥 two. We can solve this equation by
multiplying both sides by 𝑥 two. And in doing so, we can see that
the distance between the lighthouse and the second ship is 200 metres.
The total distance between the two
ships is given by the sum of the distance between this first ship and the lighthouse
and the distance between the second ship and the lighthouse. That’s 200 root three over three
plus 200. Now we were told to take root three
is equal to 1.73. We’ll replace root three with 1.73,
and we’ll calculate 200 multiplied by 1.73 over three. 200 can be written as two
multiplied by 100, and 100 multiplied by 1.73 is 173. Two multiplied by 173 is 346. So our calculation becomes 346
divided by three.
Now the digits three, four, and six
when added together do not make a multiple of three. So we’re not gonna get a
particularly nice number when we divide 346 by three. Let’s use the bus stop method and
see what we get. Three divided by three is one. Four divided by three is one
remainder one. And 16 divided by three is five
remainder one. 10 divided by three is three
remainder one, and 10 divided by three again is three remainder one.
We can see that this three is going
to reoccur, so 346 divided by three is 115.3 recurring. We can replace 200 root three over
three with 115.3 recurring in our calculation, to give us an answer of 315.3
recurring. Correct to one decimal place, the
distance between the two ships is 315.3 metres.