A vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height 6 m . At a point on the plane, the angle of elevation of the bottom and top of the flag staff are 30 ° and 45 ° respectively. Find the height of the tower. Take 3 = 1 . 73

Let A B be the tower and B C be the flagstaff. Let O A = x m , A B = y m and B C = 6 m . Here α = 30 ° and β = 45 ° In right ∆ O A B , tan 30 = A B O A 1 3 = y x ⇒ x = y 3 In right ∆ O A C tan β = A C O A ⇒ tan 45 ° = y + 6 x x = y + 6 .....(ii) From equation i and i i y 3 = y + 6 y 3 - 1 = 6 y = 6 3 - 1 Rationalising the denominator, we get, y = 6 3 + 1 3 - 1 3 + 1 = 3 3 + 1 m Put 3 = 1 . 73 , we get, y = 3 3 + 1 = 3 × 2 . 73 = 8 . 19 m Hence, the height of the tower is 8 . 19 m

As observed from the top of a 100 m high light house from the sea level, the angles of depression of two ships are 30 ° and 45 ° . If one ship is exactly behind the other on the same side of the light house, find the distance between the two ships. [ Use 3 = 1 . 732 ]

Let A B represents the lighthouse and the two ships are at point C and D respectively. In ∆ A B C , tan 45 o = A B B C ⇒ 1 = 100 BC ⇒ BC = 100 m In ∆ A B D , tan 30 o = A B B D ⇒ 1 3 = 100 BC + CD ⇒ 1 3 = 100 100 + C D ⇒ 100 + CD = 100 3 ⇒ CD = 100 3 - 100 ⇒ C D = 100 ( 3 - 1 ) ⇒ CD = 100 ( 1 . 732 - 1 ) ⇒ CD = 73 . 2 m Hence, the distance between the two ships is 73 . 2 m .

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Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 9 - Some Applications of Trigonometry>Exercise>Q 33

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An aeroplane flying at the height of $3000 m$ above the ground passes vertically above another plane at an instant when the angle of elevation of the two planes from the same point on the ground are $60 °$ and $45 °$ respectively. Then the height of the lower plane from the ground is

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Important Questions on Some Applications of Trigonometry

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Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 9 - Some Applications of Trigonometry>Exercise>Q 34

An aeroplane flying horizontally

1 km

above the ground is observed at an elevation of

60^{°}

and after

10

seconds the elevation is observed to be

30^{°}

. The uniform speed of the aeroplane (in

km / h

), is

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Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 9 - Some Applications of Trigonometry>Exercise>Q 35

If the elevation angle of a camera situated at the top of a pole from a point $20$ metre away from the base of the pole is $60 °$ , find the height of the pole.

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Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 9 - Some Applications of Trigonometry>Exercise>Q 36

The value of $h$ in the following figure is

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