Two ships are sailing in the sea on the two sides of a lighthouse.The angles of elevation of the top of the lighthouse as observed from the two ships are 30°and 45°respectively. If the lighthouse is 100 m high the distance between the two ships is

It is given that the angles of elevation of the top of the lighthouse are

{30 °}^{}

and

{45 °}^{},

observed from the two ships, and the lighthouse is

100

m in height.
According to the given data, the figure has been drawn below.

Let

AB

be the height of lighthouse and

C

and

D

be the position of the ships.
Given,

AB = 100

m and

∠ ACB = {30 °}^{}

∠ ADB = {45 °}^{}

.
In

ΔBCA

tan θ = P e r p e n d i c u l a r b a s e

\frac{AB}{AC} = \tan {30 °}^{}

\frac{AB}{AC} = \frac{1}{\sqrt{3}}

(\tan 30 ° = \frac{1}{\sqrt{3}})

\Rightarrow AC = \sqrt{3} AB

\Rightarrow AC = 100 \sqrt{3}

ΔBAD,

tan θ = P e r p e n d i c u l a r b a s e

\frac{AB}{AD} = \tan {45 °}^{}

\Rightarrow \frac{AB}{AD} =

(\tan 45 ° = 1)

\Rightarrow AD = AB = 100 m

∴ CD = AC + AD

\Rightarrow 100 \sqrt{3} + 100

\Rightarrow 100 (\sqrt{3} + 1)

\Rightarrow 100 (1.73 + 1)

(\sqrt{3} = 1.732)

\Rightarrow 273 m

Distance between the two ships is 273m.
So, the correct option is 3.

Two ships are sailing in the sea on the two sides of a lighthouse.The angles of elevation of the top of the lighthouse as observed from the two ships are ${30 °}^{}$ and ${45 °}^{}$ respectively. If the lighthouse is $100$ m high the distance between the two ships is