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

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The distance of the hill from the ship is

3

m and the height of the hill is 40 m

The distance of the hill from the ship is

103

m and the height of the hill is 40 m

The distance of the hill from the ship is

23

m and the height of the hill is 60 m

The distance of the hill from the ship is

103

m and the height of the hill is 10 m

answer is B.

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Detailed Solution

Given that a man is standing on the deck of a ship, which is 10 m above the water level, observes the angle of elevation of the top of a hill as

600

and the angle of depression of the base of hill as

300

.
We need to find the distance of hill from the ship and the height of the hill.
Question Image

From the corresponding figure, let us assume that the point of observation is at A and the height of the deck is 10 m.
Thus,

A B = C D = 10 m

Therefore, the angle of elevation of the top of hill is

∠ E A D = 600

The angle of the depression of the base of hill is

∠ D A C = 300

.
Let us assume that, in triangle

Δ A D E

A D = B C = x m

From the figure

Δ A D E

tan θ = o p p o s i t e s i d e a d j a c e n t s i d e

tan 600 = E D D A

. Use the value of trigonometric angle

tan 60 ° = 3

3 = h x h = 3 x … … … .... … … . (1)

Consider the triangle

Δ A B C

tan θ = o p p o s i t e s i d e a d j a c e n t s i d e

Thus,

tan 300 = A B B C

Use the value of the trigonometric angle

tan 30 ° = 13

13 = 10 x x = 103 … … … … … . (2)

The distance of the hill from the ship is

x = 103

.
Substitute the value of x from equation (2) in equation (1),

h = (103) 3 h = 10 × 3 h = 30 … … … … . (3)

The height of the hill is CE, i.e.,

C E = C D + D E C E = 10 + h … … … . (4)

Substitute the value of h from equation (3) in equation (4).

C E = h + 10 C E = 30 + 10 C E = 40 m

The height of the hill is 40 m.
Thus, the distance of the hill from the ship is

103

m and the height of the hill is 40 m.
Therefore, the correct option is 2.

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

Start JEE / NEET / Foundation preparation at rupees 99/day !!

Detailed Solution

Get Expert Academic Guidance – Connect with a Counselor Today!

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