Two ships are approaching a light house from opposite directions. The angles of depression of two ships from top of the light house are $30 °$ and $45 °$ . If the distance between two ships is 100m, the height of the light house is ____.

Question

Two ships are approaching a light house from opposite directions. The angles of depression of two ships from top of the light house are $30 °$ and $45 °$ . If the distance between two ships is 100m, the height of the light house is ____.

Accepted Answer

Let h be the height of the light house,

x

be the distance of the ship making angle of

45

∘ from the base of the light house.
It is given that the distance between two ships is 100m. Therefore the distance between the ship making an angle of

30 °

with the light house to the base of the light house will be 100−

x

In ΔACD, we will use the trigonometric function to get the relation between

x

and h. Therefore we get

\tan 45 ° = \frac{perpendicular}{base} = \frac{h}{x}

We know that

\tan \tan 45 °

is equal to 1. Therefore, we get
⇒1=

\frac{h}{x}

⇒h= x……………….(1)
We will now take the triangle ABC and use the trigonometric function to find the relationship between x and h.
Therefore, we get

\tan 30 ° = \frac{perpendicular}{base} = \frac{h}{100 - x}

We know that

\tan 30 °

is equal to

\frac{1}{\sqrt 3}

.
Therefore, we get
⇒

\frac{1}{\sqrt 3}

.=

\frac{h}{100 - x}

⇒

\sqrt{3} h = 100 - x

Now, substituting the value of h from the equation (1) in the above equation, we get
⇒

\sqrt{3} h

=100−h
⇒

\sqrt{3} h + h

=100
⇒h=

\frac{100}{1 + \sqrt 3}

=36.6m
Hence, the height of the lighthouse is equal to 36.6m.

Two ships are approaching a light house from opposite directions. The angles of depression of two ships from top of the light house are $30 °$ and $45 °$ . If the distance between two ships is 100m, the height of the light house is ____.

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Two ships are approaching a light house from opposite directions. The angles of depression of two ships from top of the light house are 30°and 45°. If the distance between two ships is 100m, the height of the light house is ____.

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Two ships are approaching a light house from opposite directions. The angles of depression of two ships from top of the light house are $30 °$ and $45 °$ . If the distance between two ships is 100m, the height of the light house is ____.