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**Q1. A boy whose eye level is 1.3 m from the ground, spots a balloon moving with the wind in a horizontal level at some height from the ground. The angle of elevation of the balloon from the eyes of the boy at any instant is 60Â°. After 2 seconds, the angle of elevation reduces to 30Â°. If the speed of the wind at that moment is 29âˆš3 m/s, then find the height of the balloon from ground.**

**Sol.** Let E and D be the two positions of the balloon.

Let AB be the position of the boy.

âˆ´ AB = 1.3 m

â‡’ HF = CG = 1.3 m

Also speed of the wind = 29âˆš3 m/s

Distance covered by the balloon in 2 seconds

= ED = HG = 2 Ã— 29âˆš3 m

= 58âˆš3 m

âˆ´ AG = AH + HG

= AH + 58âˆš3 m ...(1)

Now, in right Î” AEH, we have

Thus, the height of the balloon = 88.3 m.

**Q2. From a point on the ground, the angles of elevation of the bottom and the top of a tower fixed at the top of a 20 m high building are 45Â° and 60Â° respectively. Find the height of the tower. **

**Sol. **Let AB = 20 m be the building, P be the point on ground, and AC = xm be the tower.

Thus, the required height of the tower is 14.64 m

**Q3. A statue, 1.5 m tall stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 45Â° and from the same point the angle of elevation of the top of the pedestal is 30Â°. Find the height of the pedestal from the ground.**

**Sol.** Let AB be the pedestal and AB = h

Let C be the point on the ground such that

BC = x metres.

In right Î” ACB, we have:

**Q4. The angles of depression of the top and battom of an 8 m tall building from the top of a multi-storeyed building are 30Â° and 45Â°, respectively. Find the height of the multistoreyed building and the distance between the two buildings.**

**Sol. **Let the multistoreyed building be AB.

âˆ´ AB = q metres

â‡’ AD =(q âˆ’ 8) m [âˆµ BD = 8 m]

Let EC be the small building.

Now, in right Î” ABC, we have:

âˆ´ Distance between the two buildings = 18.928 m

Height of the multi-storeyed building = 18.928 m.

**Q5. From the top of a building 60 m high, the angles of depression of the top and bottom of a vertical lamp post are observed to be 30Â° and 60Â° respectively. Find:**

**(i) The horizontal distance between the building and the lamp post. (ii) The height of the lamp post.**

**[Take âˆš3 = 1.732]**

**Sol. **In the figure, let CE be the building and AB be the lamp post

âˆ´ CE = 60 m

= 60 m âˆ’ 20

m = 40 m.

Also, the distances between the lamp post and the building

= 20âˆš3 m = 20 Ã— 1.732 m [âˆµ 3 = 1.732]

= 34.64 m

**Q6. The angle of elevation of a cloud from a point h meters above the surface of a lake is q and the angle of depression of its reflection in the lake is Ï†. Prove that the height of the clouds above the lake is **

**Sol.** Let P be the cloud and Q be its reflection in the lake. As shown in the figure, let A be the point of observation such that AB = h

Let the height of the cloud above the lake = x

Let AL = d

From rt Î”PLA, tan Î¸ =

**Q7. From a point 100 m above a lake, the angle of elevation of a stationary helicopter is 30Â° and the angle of depression of reflection of the helicopter in the lake is 60Â°. Find the height of the helicopter.**

**Sol. **In the figure, A is the stationary helicopter and F is its reflection in the lake.

In right Î” AED, we have:

Thus, the height of the stationary helicopter = 200 m.

**Q8. The angle of elevation of an aeroplane from a point on the ground is 60Â°. After a flight of 15 seconds, the angle of elevation changes to 30Â°. If the aeroplane is flying at a constant height of 1500âˆš3 m, find the speed of the aeroplane.**

**Sol. **In the figure, let E and C be the two locations of the aeroplane.

Height BC = ED

= 1500âˆš3 m

In right Î” ABC, we have:

âˆ´ Speed of the aeroplane

= 200 m/s.

**Q9. A spherical balloon of radius r subtends an angle q at the eye of the observer. If the angle of elevation of its centre is Ï†, find the heights of centre of the balloon.**

**Sol. **In the figure, let O be the centre of the balloon, and A be the eye of the observer. r be the radius.

âˆ´ OP = r and PAQ = Î¸

Also, âˆ OAB = Ï†

Let the height of the centre of the balloon be â€˜hâ€™ â‡’ OB = h.

InÎ”OAP, âˆ OPA = 90Â°

**Q10. As observered from the top of a light house, 100 m high above sea level, the angle of depression of a ship sailing directly towards it, changes from 30Â° to 60Â°. Determine the distances travelled by the ship during the period of observation. [Useâˆš3 = 1.732]**

**Sol. **Let A represents the position of the observer such that AB = 100 m

âˆ´ In right Î” ABC, we have

**Q11. From the top of a tower, 100 m high, a man observes two cars on the opposite sides of the tower with angles of depression 30Â° and 45Â° respectively. Find the distances between the cars. [useâˆš3 = 1.73]**

**Sol. **Let 'P' be the position of the observer.

A and B represent the cars. Height of the tower PQ = 100 m In rt Î” PQA,

Thus, distance between the cars A and B :

**Q12. The angles of elevation and depression of the top and the bottom of a tower from the top of a building, 60m high, are 30Â° and 60Â° respectively. Find the difference between the heights of the building and the tower and the distance between them.**

**Sol. **Let AB is building = 60 m and DC is the tower

Substituting the value of x from (2) in (1), we have :

â‡’ Difference between the heights of building and tower = 20 m

Distance between the tower and building

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