Two ships are sailing on the sea on either side of the lighthouse The angle of the depression of the - Maths - Some Applications of Trigonometry

Question

Two ships are sailing on the sea on either side of the
lighthouse The angle of the depression of the ship as observed from
the lighthouse are 60o and 45o respectively
If the distance between the ship is 200(root 3 +1)/root 3
metre,find the height of the lighthouse

Ashim Sharma · Accepted Answer

Let A and B be the positions of the ships on either side of the light house.

AB = $n n 200 n n (n \frac{n}{n \sqrt{n n 3 n n} n + n 1 n} n) n n m n n$

Let BD = x m

∴ AD = AB – BD $n n = n n [n 200 n n (n \frac{n}{n \sqrt{n n 3 n n} n + n 1 n} n) n - n x n n] n n m n n$

Let the height of the light house be h m.

In ΔBCD,

$n n \tan n n 60 n ° n = n \frac{n}{n CD n} n n ∴ n \sqrt{n n 3 n n} n = n \frac{n}{n h n} n n h n = n \sqrt{n n 3 n n} n x n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n ... n (n 1 n) n n$

In ΔACD,

$n n \tan n n 45 n ° n = n \frac{n}{n CD n} n n ∴ n 1 n = n \frac{n}{n h n} n n \Rightarrow n h n = n n 200 n n (n \frac{n}{n \sqrt{n n 3 n n} n + n 1 n} n) n n - n x n n$

From (1) and (2) we get

$n n \sqrt{n n 3 n n} n x n = n 200 n n (n \frac{n}{n \sqrt{n n 3 n n} n + n 1 n} n) n n - n x n n \Rightarrow n \sqrt{n n 3 n n} n x n + n x n = n 200 n n (n \frac{n}{n \sqrt{n n 3 n n} n + n 1 n} n) n n n \Rightarrow n n (n \sqrt{n n 3 n n} n + n 1 n) n x n = n 200 n n (n \frac{n}{n \sqrt{n n 3 n n} n + n 1 n} n) n n n \Rightarrow n x n = n \frac{n}{n 200 n} n n n n n m n n n n n$

From (1) we have

$n n h n = n \sqrt{n n 3 n n} n n n n n \times n n n n \frac{n}{n 200 n} n n m= 200m n n$

Thus, the height of light house is 200 m.

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Two ships are sailing on the sea on either side of the lighthouse.The angle of the depression of the ship as observed from the lighthouse are 60o and 45o respectively.If the distance between the ship is 200(root 3 +1)/root 3 metre,find the height of the lighthouse.

Two ships are sailing on the sea on either side of the lighthouse.The angle of the depression of the ship as observed from the lighthouse are 60^o and 45^o respectively.If the distance between the ship is 200(root 3 +1)/root 3 metre,find the height of the lighthouse.