Question

The angles of depression of two ships from the top of a lighthouse are ${45}^o$ and ${30}^o$ towards east. If the ships are 100 m apart, the height of the lighthouse is _______.

• A. $\frac{50}{\sqrt{3}+1}m$
• B. $\frac{50}{\sqrt{3}-1}m$
• C. $50(\sqrt{3}+1)m$
• D. None of these

Answer 1

The correct option is C $50(\sqrt{3}+1)m$

Say $AB=h$ (height of the tower)

In $△AB{S}_1$,

$\tan 45°=\frac{AB}{B{S}_1}=>B{S}_1=AB=>B{S}_1=h$

Similarly in $△AB{S}_2$,

$\tan 30°=\frac{AB}{B{S}_2}=\frac{AB}{(B{S}_1+S_1{S}_2)}=>\frac{1}{\sqrt{3}}=\frac{h}{h+100}=>100+h=\sqrt{3}h=>h=\frac{100}{\sqrt{3}-1}$

$=>h=50\times (\sqrt{3}+1)$ m.