CBSE Class 10 Maths (Standard) 2025 · Paper 30/3/1 · Question 37

Some Applications of Trigonometry 4 marks Medium

A lighthouse stands tall on a cliff by the sea, watching over ships that pass by. One day a ship is seen approaching the shore and from the top of the lighthouse, the angles of depression of the ship are observed to be

3 0^{\circ}

and

4 5^{\circ}

as it moves from point

P

to point

Q

. The height of the lighthouse is

50

metres. Based on the information given above, answer the following questions:

Right triangle showing lighthouse and ship positions P, Q with angles of depression.

(i)Find the distance of the ship from the base of the lighthouse when it is at point $Q$ , where the angle of depression is $4 5^{\circ}$ .[1]
(ii)Find the measures of $∠ P B A$ and $∠ QB A$ .[1]
(iii)(a)Find the distance travelled by the ship between points $P$ and $Q$ .[2]
(iii)(b) (OR)If the ship continues moving towards the shore and takes $10$ minutes to travel from $Q$ to $A$ , calculate the speed of the ship in km/h, from $Q$ to $A$ .[2]

Answer

(i) At

Q

the ship is

A Q = 50

m from the base of the lighthouse; (ii)

∠ P B A = 6 0^{\circ}

and

∠ QB A = 4 5^{\circ}

; (iii)(a) The ship travels

P Q = 50 (3 - 1) \approx 36.6

m; (iii)(b) Its speed from

Q

A

0.3

km/h.

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