A man standing on the deck of a ship, which is 10m above the water level, observes the angle of - Maths - Some Applications of Trigonometry

Question

A man standing on the deck of a ship, which is 10m above the
water level, observes the angle of elevation of the top of a hill
as 60 degree and the angle of depression of the base of the hill as
30 degree Find the distance of the hill fom the ship and the height
of the hill

Vimala · Accepted Answer

Let us observe the following figure

The point of observation is at A The height of the deck is 10 meters Thus, AB = CD = 10 meters The top and bottom of a hill is E and C Therefore the angles of elevation and the depression are $∠ E A D = 60^{\circ} a n d ∠ D A C = 30^{\circ}$ Let AD = BC = x meters Consider the triangle ADE Thus, $\tan 60^{\circ} = \frac{E D}{D A} \sqrt{3} = \frac{h}{x} h = x \sqrt{3} (1)$ ; Now consider the triangle ABC $\tan 30^{\circ} = \frac{A B}{B C} \frac{1}{\sqrt{3}} = \frac{10}{B C} B C = 10 \sqrt{3} x = 10 \sqrt{3} (2)$ Substitute the value of x from equation (2) in equation (1), we have $h = (10 \sqrt{3}) \sqrt{3} (3)$ Therefore h = 30 meters (4) The height of the hill is CE That is, $C E = C D + D E = 10 + h (5)$ Substitute the value of h from equation (4) in equation (5), we hve $C E = h + 10 m e t e r s = 30 + 10 = 40 m e t e r s$ Thus, the height of the hill is 40 meters and the distance of the hill from the ship is $10 \sqrt{3} m e t e r s$

A man standing on the deck of a ship, which is 10m above the water level, observes the angle of elevation of the top of a hill as 60 degree and the angle of depression of the base of the hill as 30 degree. Find the distance of the hill fom the ship and the height of the hill.