Video: CBSE Class X • Pack 5 • 2014 • Question 19

CBSE Class X • Pack 5 • 2014 • Question 19

05:23

Video Transcript

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 1.73.

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 degrees.

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 angle.

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.

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