Question

As observed from the top of a lighthouse, 100 m above the sea level, the angle of depression of a ship, sailing directly towards it, changes from 30° to 45°. Determine the distance travelled by the ship during the period of observation.

Answer 1

Hint: Solve by representing the text with a diagram, then use trigonometric formulas to find the relation between the given variables. Then calculate the distance travelled by the ship during the period of observation.

Complete step-by-step answer:

In the given figure, AB is the lighthouse and is 100 m in height. The angle of depression of the ship at C is 30° and the angle of depression of the ship at D is 45°.
We know that alternate angles between two parallel lines are equal. Hence, the angle ACB is equal to 30° and the angle ADB is equal to 45°.
We can use the tangent of the angle to find the distance of travel during the period of observation.
In triangle ABC, the tangent of the angle C is the ratio of side AB to side BC.
\[\tan C = \dfrac{{AB}}{{BC}}\]
Angle C is 30° and AB is 100m, then, we can find the value of the side BC.
\[\tan 30^\circ = \dfrac{{100}}{{BC}}\]
We know that the value of \[\tan 30^\circ \] is \[\dfrac{1}{{\sqrt 3 }}\], then, we have:
\[\dfrac{1}{{\sqrt 3 }} = \dfrac{{100}}{{BC}}\]
Cross multiplying, we have:
\[BC = 100\sqrt 3 m..........(1)\]
Now, in triangle ABD, the tangent of the angle D is the ratio of side AB to side BD.
\[\tan D = \dfrac{{AB}}{{BD}}\]
Angle D is 45° and AB is 100m, then, we can find the value of the side BD.
\[\tan 45^\circ = \dfrac{{100}}{{BD}}\]
We know that the value of \[\tan 45^\circ \] is 1, then, we have:
\[1 = \dfrac{{100}}{{BD}}\]
Cross multiplying, we have:
\[BD = 100m..........(2)\]
From equations (1) and (2), we can find the value of CD.
\[CD = BC - BD\]
\[CD = 100\sqrt 3 - 100\]
\[CD = 100(\sqrt 3 - 1)m\]
Hence, the answer is \[100(\sqrt 3 - 1)m\].

Note: You may think that the angle of depression is the angle made by the observer’s line of sight with the vertical line but it is wrong. It is the angle made between the line of sight and the horizontal line.