Question

From the top of a lighthouse 60-metre-high, with its base at the sea level, the angle of depression of a boat is ${{30}^{\circ }}$. The distance of the boat from the foot of the light house is?
(a) $60\sqrt{3}$ metres
(b) $\dfrac{60}{\sqrt{3}}$ metres
(c) 60 metres
(d) $30\sqrt{2}$ metres

Answer 1

Hint: We start solving the problem by drawing the figure representing the given information. We then find the angle of elevation of the boat from the top of light house as we know that the sum of angle of elevation and angle of depression at a point is ${{90}^{\circ }}$. We then recall the fact that the tangent of an angle in a right-angled triangle is equal to the ratio of the opposite side to the adjacent side. We apply this property to the obtained angle of elevation and make necessary calculations to get the required distance.

Complete step by step answer:
According to the problem, we are given that the angle of depression of a boat from the top of a lighthouse 60-metre-high with its base at the sea level is ${{30}^{\circ }}$. We need to find the distance of the boat from the foot of the light house.
Let us draw the figure representing the given information.

Let us assume AB be the light tower and C is the point where the boat is present. We have given that the angle of depression is ${{30}^{\circ }}$.
From the figure, we have $\alpha +{{30}^{\circ }}={{90}^{\circ }}$.
$\Rightarrow \alpha ={{60}^{\circ }}$ ---(1).
From the figure, we can see that AC is the distance between the boat and the foot of the light tower.
We know that the tangent of an angle in a right-angled triangle is equal to the ratio of the opposite side to the adjacent side.
So, we get $\tan {{60}^{\circ }}=\dfrac{AC}{AB}$.
$\Rightarrow \sqrt{3}=\dfrac{AC}{60}$.
$\Rightarrow AC=60\sqrt{3}m$.
So, we have found the distance of the boat from the foot of the lighthouse as $60\sqrt{3}m$.

So, the correct answer is “Option a”.

Note: We can also find the angle of elevation of the top of the lighthouse from the boat using the fact that the sum of angles in a triangle is ${{180}^{\circ }}$. Whenever we get this type of problems, we first draw the figure representing the given information as it gives the better view of it which leads to the solution. We can also find the distance of line of sight from the top of the lighthouse to the boat (BC) using the Pythagoras theorem.