As observed from the top of a $ 75 \mathrm{~m} $ tall lighthouse, the angles of depression of two ships are $ 30^{\circ} $ and $ 45^{\circ} $. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.
Given:
As observed from the top of a \( 75 \mathrm{~m} \) tall lighthouse, the angles of depression of two ships are \( 30^{\circ} \) and \( 45^{\circ} \).
One ship is exactly behind the other on the same side of the lighthouse.
To do:
We have to find the distance between the two ships.
Solution:
Let $AB$ be the height of the tall lighthouse and $C, D$ are the two ships one behind the other.
From the figure,
$\mathrm{AB}=75 \mathrm{~m}, \angle \mathrm{BCA}=30^{\circ}, \angle \mathrm{BDA}=45^{\circ}$
Let the distance between ship $D$ and the lighthouse be $\mathrm{DA}=x \mathrm{~m}$ and the distance between the two ships be $\mathrm{CD}=y \mathrm{~m}$.
This implies,
$\mathrm{CA}=x+y \mathrm{~m}$
We know that,
$\tan \theta=\frac{\text { Opposite }}{\text { Adjacent }}$
$=\frac{\text { AB }}{DA}$
$\Rightarrow \tan 45^{\circ}=\frac{75}{x}$
$\Rightarrow 1=\frac{75}{x}$
$\Rightarrow x=75 \mathrm{~m}$.........(i)
Similarly,
$\tan \theta=\frac{\text { Opposite }}{\text { Adjacent }}$
$=\frac{\text { BA }}{CA}$
$\Rightarrow \tan 30^{\circ}=\frac{75}{x+y}$
$\Rightarrow \frac{1}{\sqrt3}=\frac{75}{75+y}$ [From (i)]
$\Rightarrow 75+y=75\sqrt3 \mathrm{~m}$
$\Rightarrow y=75(\sqrt3-1) \mathrm{~m}$
Therefore, the distance between the two ships is $75(\sqrt3-1) \mathrm{~m}$.
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