UPSC NDA Math Model Paper 6

From the top of a lighthouse, 70 m high with its base at sea level, the angle of depression of a boat is 15°.
Consider the base is 'b'. tan‌θ=

70

x

⇒tan‌15°=

70

x

.......(1)
⇒tan(45°−30°)=

70

x

We know that,
tan(A−B)=

tan‌A−tan‌B

1+tan‌A.tan‌B

⇒tan(45°−30°)=

tan‌45°−tan‌30°

1+tan‌45°.tan‌30°

⇒tan(15°)=

1− 1 √3

1+1. 1 √3

⇒tan(15°)=

√3−1

√3+1

Bt Rationalisation, we get
⇒tan‌15°=2−√3
From equation (1), we get
⇒

70

x

=2−√3
⇒x=

70

2−√3

⇒x=

70

2−√3

×

2+√3

2+√3

⇒x=70(2+√3)m
Hence, The distance of the boat from the foot of the lighthouse is 70(2+√3)m