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#### Question 1:

A tower stands vertically on the ground. From a point on the ground which is 20 m away from the foot of the tower, the angle of elevation of its top is found to be 60°. Find the height of the tower.

Let $AB$ be the tower standing vertically on the ground and O be the position of the observer.
We now have:
and ∠$AOB$
Let:
m Now, in the right ∆$OAB$, we have:
= $\sqrt{3}$

⇒

⇒  =

Hence, the height of the pole is 34.64 m.

#### Question 2:

A kite is flying at a height of 75 m from the level ground, attached to a string inclined at a 60° to the horizontal. Find the length of the string assuming that there is no slack in it.

Let  be the horizontal ground and $A$ be the position of the kite.
Also, let O be the position of the observer and $OA$ be the thread.
Now, draw ⊥ $OX$.
We have:
${60}^{o}$m and ∠
Height of the kite from the ground = $AB$ = 75 m
Length of the string,  m In the right ∆$OBA$, we have:

⇒  m
Hence, the length of the string is $86.6$ m.

#### Question 3:

An observer 1.5 m tall is 30 m away from a chimney. The angle of elevation of the top of the chimney from his eye is 60°.
Find the height of the chimney.                                                                                                            [CBSE 2013C] Let CE and AD be the heights of the observer and the chimney, respectively.

We have,

So, the height of the chimney is 53.46 m (approx.).

#### Question 4:

The angles of elevation of the top of a tower from two points at distances of 5 metres and 20 metres from the base of the tower and in the same straight line with it, are complementary. Find the height of the tower.                                                                               [CBSE 2014] Let the height of the tower be AB.

We have,

So, the height of the tower is 10 m.

#### Question 5:

The angle of elevation of the top of a tower at a distance of 120 m from a point A on the ground is 45°. If the angle of elevation of the top of a flagstaff fixed at the top of the tower, at A is 60°, then find the height of the flagstaff. [Use $\sqrt{3}$ = 1.732]                                  [CBSE 2014] Let BC and CD be the heights of the tower and the flagstaff, respectively.

We have,

So, the height of the flagstaff is 87.8 m.

#### Question 6:

From a point on the ground 40 m away from the foot of a tower, the angle of elevation of the top of the tower is 30°. The angle of elevation of the top of a water tank (on the top of the tower) is 45°. Find (i) the height of the tower, (ii) the depth of the tank. Let BC be the tower and CD be the water tank.

#### Question 7:

A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height 6 m. At a point on the plane, the angle of elevation of the bottom of the flagstaff is 30° and that of the top of the flagstaff is 60°. Find the height of the tower.
[Use $\sqrt{3}$ = 1.732]                                                                                                                                                                     [CBSE 2011] Let AB be the tower and BC be the flagstaff.

We have,

So, the height of the tower is 3 m.

#### Question 8:

A statue 1.46 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 60° and from the same point, the angle of elevation of the top of the pedestal is 45°. Find the height of the pedestal.

Let $AC$ be the pedestal and $BC$ be the statue such that  m.
We have:
and ∠
Let:
m and  m In the right ∆$ADC$, we have:

⇒
⇒
Or,

Now, in the right ∆$ADB$, we have:

⇒

On putting  in the above equation, we get:

⇒
⇒
⇒ m

Hence, the height of the pedestal is 2 m.

#### Question 9:

The angle of elevation of the top of an unfinished  tower at a distance of 75 m form its base is 30°. How much higher must the tower be raised so that the angle of elevation of its top at the same point may be 60°?

Let $AB$ be the unfinished tower, $AC$ be the raised tower and O be the point of observation.
We have:
m, ∠ and ∠
Let  m such that  m. In ∆AOB, we have:

⇒  m =  m
In ∆$AOC$, we have:
$=\sqrt{3}$

⇒
m

∴ Required height =  m = 86.6 m

#### Question 10:

On a horizontal plane there is a vertical tower with a flagpole on the top of the tower. At a point, 9 metres away from the foot of the tower, the angle of elevation of the top of bottom of the flagpole are 60° and 30° respectively. Find the height of the tower and the flagpole mounted on it.

Let $OX$ be the horizontal plane, be the tower and $CD$ be the vertical flagpole.
We have:
m, ∠ and ∠
Let:
m and  m In the right ∆$ABD$, we have:

⇒ $\frac{h}{9}=\frac{1}{\sqrt{3}}$
⇒  m
Now, in the right ∆$ABC$, we have:

⇒
⇒

By putting  in the above equation, we get:

⇒
⇒

Thus, we have:
Height of the flagpole = 10.4 m
Height of the tower = 5.19 m

#### Question 11:

Two poles of equal heights are standing opposite to each other on either side of the road which is 80 m wide. From a point P between them on
the road, the angle of elevation of the top of one pole is 60° and the angle of depression from the top of another pole at P is 30°. Find the
height of each pole and distances of the point P from the poles.                                                                                                         [CBSE 2015] Let AB and CD be the equal poles; and BD be the width of the road.

We have,

Hence, the height of each pole is 20$\sqrt{3}$ m and point P is at a distance of 20 m from left pole and 60 m from right pole.

#### Question 12:

Two men are on opposite sides of a tower. they measure the angles of elevation of the top of the tower as 30° and 45° respectively. If the height of the tower is 50 metres, find the distance between the two men.

Let $CD$ be the tower and  be the positions of the two men standing on the opposite sides. Thus, we have:
, ∠ and  m
Let  m and  m such that  m. In the right ∆$DBC$, we have:

⇒
⇒  m

In the right ∆$ACD$, we have:

⇒
⇒
On putting  in the above equation, we get:

⇒  m

∴ Distance between the two men =  m

#### Question 13:

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 30° and 45°, respectively. Find the distance between the cars. [Take $\sqrt{3}$= 1.73]                                                                          [CBSE 2011] Let PQ be the tower.

We have,

So, the distance between the cars is 273 m.

#### Question 14:

A straight highway leads to the foot of a tower. A man standing on the top of the tower observes a car at an angle of depression of 30°, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be 60°. Find the time taken by the car to reach the foot of the tower form this point. Let PQ be the tower.

So, the time taken to reach the foot of the tower from the given point is 3 seconds.

#### Question 15:

A TV tower stands vertically on a bank of canal. From a point on other bank directly opposite the tower, the angle of elevation of the top of the tower is 60°. From another point 20 m away from this point on the line joining this point to the foot of the tower, the angle of elevation of the top of the tower is 30°. Find the height of the tower and the width of the canal. Let PQ=h m be the height of the TV tower and BQ=x m be the width of the canal.

We have,

So, the height of the TV tower is  and the width of the canal is 10 m.

#### Question 16:

The angle of elevation of the top of a building from the foot of a tower is 30°. The angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 60 m high, then find the height of the building.                                                                          [CBSE 2013] Let AB be the building and PQ be the tower.

We have,

So, the height of the building is 20 m.

#### Question 17:

The horizontal distance between two towers is 60 metres. The angle of depression of the top of the first tower when seen from the top of the second tower is 30°. If the height of the second tower is 90 metres, find the height of the first tower.

Let $DE$ be the first tower and $AB$ be the second tower.
Now,  m and  m such that  m and ∠.
Let  m such that  m and  m. In the right ∆$BCE$, we have:

⇒
⇒
⇒
⇒  =  m
∴ Height of the first tower =  m

#### Question 18:

The angle of elevation of the top of a chimney from the foot of a tower is 60° and the angle of depression of the foot of the chimney from the top of the tower is 30°. If the height of the tower is 40 metres, then find the height of the chimney.

According to pollution control norms, the minimum height of a smoke emitting chimney should be 100 metres. State if the height of the above mentioned chimney meets the pollution norms. What value is discussed in this question?                                                   [CBSE 2014] Let PQ be the chimney and AB be the tower.

We have,

So, the height of the chimney is 120 m.

Hence, the height of the chimney meets the pollution norms.

In this question, management of air pollution has been shown.
a

#### Question 19:

From the top of a 7-metre-high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 45°. Determine the height of the tower. [Use $\sqrt{3}$ = 1.732]                                                                                                      [CBSE 2013C] Let AB be the 7-m high building and CD be the cable tower.

We have,

So, the height of the tower is 19.12 m.

#### Question 20:

The angle of depression from the top of a tower of a point A on the ground is 30°. On moving a distance of 20 metres from the point A towards the foot of the tower to a point B, the angle of elevation of the top of the tower from the point B is 60°. Find the height of the tower and its distance from the point A.                                                                                                                                     [CBSE 2012] Let PQ be the tower.

We have,

So, the height of the tower is 17.32 m and its distance from the point A is 30 m.

#### Question 21:

The angle of elevation of the top of a vertical tower from a point on the ground is 60°. From another point 10 m vertically above the first, its angle of elevation is 30°. Find the height of the tower.                                                                                                           [CBSE 2011] Let PQ be the tower.

We have,

So, the height of the tower is 15 m.

#### Question 22:

The angles of depression of the top and bottom of a tower as seen from the top of a 60$\sqrt{3}$-m-high cliff are 45° and 60°, respectively. Find the height of the tower.                                                                                                                                                                  [CBSE 2012] Let AD be the tower and BC be the cliff.

We have,

So, the height of the tower is 43.92 m.

#### Question 23:

A man on the deck of a ship, 16 m above water level observes that the angles of elevation and depression respectively of the top and bottom of a cliff are 60° and 30°. Calculate the distance of the cliff from the ship and height of the cliff.

Let $AB$ be the deck of the ship above the water level and $DE$ be the cliff.
Now,
m such that  m and ∠​ and ∠.
If AD = x m and  m, then  m. In the right ∆$BAD$, we have:

⇒

⇒   m

In the right ∆$EBC$, we have:

⇒
⇒
⇒         [∵ ]
m

∴ Distance of the cliff from the deck of the ship =  m
And,
Height of the cliff =  m

#### Question 24:

The angle of elevation of the top Q of a vertical tower PQ from a point X on the ground is 60°. At a point Y, 40 m vertically above X, the angle of elevation is 45°. Find the height of tower PQ. [Take $\sqrt{3}$ = 1.73]                                                                                  [CBSE 2003C] We have,

So, the height of the tower PQ is 94.6 m.

#### Question 25:

The angle of elevation of an aeroplane from a point on the ground is 45°. After flying for 15 seconds, the elevation changes to 30°. If the aeroplane is flying at a height of 2500 metres, find the speed of the aeroplane. Let the height of flying of the aeroplane be PQ = BC and point A be the point of observation.

We have,

So, the speed of the aeroplane is 122 m/s or 439.2 km/h.

#### Question 26:

The angle of elevation of the top of a tower from a point on the same level as the foot of the tower is 30° On advancing 150 m towards the foot of the tower, the angle of elevation becomes 60°. Show that the height of the tower is 129.9 metres.

Let $AB$ be the tower.
We have:
m, ∠ and ∠
Let:
m  and  m In the right ∆$ABD$, we have:

⇒
⇒
Now, in the right ∆$ACB$, we have:

⇒
⇒

On putting  in the above equation, we get:

⇒
⇒
⇒  m
Hence, the height of the tower is 129.9 m.

#### Question 27:

As observed form the top of a lighthouse, 100 m above sea level, the angle of depression of a ship, sailing directly towards it, changes from 30° to 60°. Determine the distance travelled by the ship during the period of observation.

Let $OA$ be the lighthouse and B and C be the two positions of the ship.
Thus, we have:
m, ∠ and  ∠ Let:
m and  m
In the right ∆$OAC$, we have:

⇒ m
Now, in the right ∆$OBA$, we have:

⇒

On putting  in the above equation, we get:
m

∴ Distance travelled by the ship during the period of observation =  m

#### Question 28:

From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are 30° and 45° respectively. If the bridge is at a height of 2.5 m from the banks, find width of the river. Let $A$ and $B$ be two points on the banks on the opposite side of the river and $P$ be the point on the bridge at a height of 2.5 m.
Thus, we have:
In the right ∆$APD$, we have:

⇒
⇒  m
In the right ∆$PDB$, we have:

⇒
⇒  m

∴ Width of the river =  m

#### Question 29:

The angles of elevation of the top of a tower from two points at distances of 4 m and 9 m from the base of the tower and in the same straight line with it are complementary. Show that the height of the tower is 6 metres.

Let $AB$ be the tower and  be two points such that  m  and  m.
Let:
m, ∠ and ∠ In the right ∆BCA, we have:

In the right ∆BDA, we have:

Multiplying equations (1) and (2), we get:

⇒ 36 = h2
h = ±6

Height of a tower cannot be negative.
∴ Height of the tower = 6 m

#### Question 30:

A ladder of length 6 metres makes an angle of 45° with the floor while leaning against one wall of a room. If the foot of the ladder is kept fixed on the floor and it is made to lean against the opposite wall of the room, it makes an angle of 60° with the floor. Find the distance between two walls of the room.                                                                                                                                                     [CBSE 2011] Let AB and CD be the two opposite walls of the room and the foot of the ladder be fixed at the point O on the ground.

We have,

So, the distance between two walls of the room is 7.24 m.

#### Question 31:

From the top of a vertical tower, the angles of depression of two cars in the same straight line with the base of the tower, at an instant are found to be 45° and 60°. If the cars are 100 m apart and are on the same side of the tower, find the height of the tower.         [CBSE 2011] Let OP be the tower and points A and B be the positions of the cars.

We have,

So, the height of the tower is 236.6 m.

Disclaimer: The answer given in the texbook is incorrect. The same has been rectified above.

#### Question 32:

An electrician has to repair an electric fault on a pole of height 4 metres. He needs to reach a point 1 metre below the top of the pole to undertake the repair work. What should be the length of the ladder that he should use, which when inclined at an angle of 60° to the horizontal would enable him to reach the required position? [Use $\sqrt{3}$ = 1.73] Let AC be the pole and BD be the ladder.

We have,

So, he should use 3.46 m long ladder to reach the required position.

#### Question 33:

From the top of a building AB, 60 m high, the angles of depression of the top and bottom of a vertical lamp post CD are observed to be 30° and 60°, respectively. Find
(i) the horizontal distance between AB and CD,
(ii) the height of the lamp post,
(iii) the difference between the heights of the building and the lamp post.                                                                                    [CBSE 2009] We have,

#### Question 34:

A man observes a car from the top of a tower, which is moving towards the tower with a uniform speed. If the angle of depression of the car changes from 30° to 45° in 12 minutes, find the time taken by the car now to reach the tower.          [CBSE 2017]

Suppose AB be the tower of height h meters. Let C be the initial position of the car and let after 12 minutes the car be at D. It is given that the angles of depression at C and D are 30º  and 45º respectively.
Let the speed of the car be v meter per minute. Then,
CD = distance travelled by the car in 12 minutes
CD = 12v meters Suppose the car takes t minutes to reach the tower AB from D. Then DA = vt meters.

Substituting the value of h from equation (i) in equation (ii), we get

#### Question 35:

An aeroplane is flying at a height of 300 m above the ground. Flying at this height the angles of depression from the aeroplane of two points on both banks of a river in opposite directions are 45° ad 60° respectively. Find the width of the river.            [CBSE 2017]

Let CD be the height of the aeroplane above the river at some instant. Suppose A and B be two points on both banks of the river in opposite directions. Height of the aeroplane above the river, CD = 300 m
Now,
$\angle$CAD = $\angle$ADX = 60º       (Alternate angles)
$\angle$CBD = $\angle$BDY = 45º        (Alternate angles)
In right ∆ACD,

In right ∆BCD,

∴ Width of the river, AB
= BC + AC

Thus, the width of the river is 473 m.

#### Question 36:

From a point on the ground the angles of elevation of the bottom and top of a communication tower fixed on the top of a 20-m-high building are 45° and 60° respectively. Find the height of the tower.              [CBSE 2017] Let BC be the 20 m high building and AB be the communication tower of height h fixed on top of the building. Let D be a point on ground such that CD = x m and angles of elevation made from this point to top and bottom of tower are $45°$ and $60°$.
In

Also, in

#### Question 37:

From the top of a hill, the angles of depression of two consecutive kilometre stones due east are found to be 45° and 30° respectively. Find the height of the hill.            [CBSE 2017]

Let PQ be the hill of height h km. Let R and S be two consecutive kilometre stones, so the distance between them is 1 km.
Let QR = x km. From equation (i) and (ii) we get,

Hence, the height of the hill is 1.365 km.

#### Question 38:

If at some time of the day the ratio of the height of a vertically standing pole to the length of its shadow on the ground is $\sqrt{3}:1$ then find the angle of elevation of the sun at that time.             [CBSE 2017] Let AB be the vertically standing pole of height h units and CB be the length of its shadow of s units.
Since, the ratio of length of pole and its shadow at some time of day is given to be $\sqrt{3}:1.$
$⇒\frac{\mathrm{AB}}{\mathrm{BC}}=\frac{\sqrt{3}}{1}.$
In

#### Question 1:

Choose the correct answer of the following question:

If the height of a vertical pole is equal to the length of its shadow on the ground, the angle of elevation of the sun is

(a) 0°                         (b) 30°                         (c) 45°                         (d) 60°                                                                            [CBSE 2014] Let AB represents the vertical pole and BC represents the shadow on the ground and θ represents angle of elevation the sun.

Hence, the correct answer is option (c).

#### Question 2:

Choose the correct answer of the following question:

If the height of a vertical pole is $\sqrt{3}$ times the length of its shadow on the ground, then the angle of elevation of the sun at that time is

(a) 30°                    (b) 45°                     (c) 60°                     (d) 75°                                                                                [CBSE 2012, 14] Here, AO be the pole; BO be its shadow and $\theta$ be the angle of elevation of the sun.

Hence, the correct answer is option (c).

#### Question 3:

If the length of the shadow of a tower is $\sqrt{3}$ times its height then the angle of elevation of the sun is

(a) 45°
(b) 30°
(c) 60°
(d) 90°

(b) 30°

Let $AB$ be the pole and $BC$ be its shadow. Let  and   such that  (given) and $\theta$ be the angle of elevation.
From ∆$ABC$, we have:

⇒
⇒
⇒

Hence, the angle of elevation is ${30}^{\mathrm{o}}$.

#### Question 4:

Choose the correct answer of the following question:

If a pole 12 m high casts a shadow $4\sqrt{3}$ m long on the ground, then the sun's elevation is

(a) 60°                    (b) 45°                    (c) 30°                    (d) 90°                                                      [CBSE 2013C] Let AB be the pole, BC be its shadow and $\theta$ be the sun's elevation.

Hence, the correct answer is option (a).

#### Question 5:

Choose the correct answer of the following question:

The shadow of a 5-m-long stick is 2 m long. At the same time, the length of the shadow of a 12.5-m-high tree is

(a) 3 m                    (b) 3.5m                   (c) 4.5 m                    (d) 5 m                                                                      [CBSE 2011] Let AB be a stick and BC be its shadow; and PQ be the tree and QR be its shadow.

We have,

Hence, the correct answer is option (d).

#### Question 6:

Choose the correct answer of the following question:

A ladder makes an angle of 60° with the ground when placed against a wall. If the foot of the ladder is 2 m away from the wall, then the length of the ladder is

(a) $\frac{4}{\sqrt{3}}$ m                   (b) $4\sqrt{3}$ m                   (c) $2\sqrt{2}$ m                   (d) 4 m                                        [CBSE 2014] Let AB be the wall and AC be the ladder.

We have,

Hence, the correct answer is option (d).

#### Question 7:

Choose the correct answer of the following question:

A ladder 15 m long makes an angle of 60° with the wall. Find the height of the point, where the ladder touches the wall.

(a) $15\sqrt{3}$ m

(b) $\frac{15\sqrt{3}}{2}$m

(c) $\frac{15}{2}$ m

(d) 15 m Let AB be the wall and AC be the ladder.

We have,

Hence, the correct answer is option (c).

#### Question 8:

Choose the correct answer of the following question:

From a point on the ground, 30 m away from the foot of a tower, the angle of elevation of the top of the tower is 30°. The height of the
tower is

(a) 30 m                    (b) $10\sqrt{3}$ m                    (c) 10 m                    (d) $30\sqrt{3}$ m                                                                       [CBSE 2014] Let AB be the tower and point C be the point of observation on the ground.

We have,

Hence, the correct answer is option (b).

#### Question 9:

Choose the correct answer of the following question:

The angle of depression of a car parked on the road from the top of a 150-m-high tower is 30°. The distance of the car from the tower is

(a) $50\sqrt{3}$ m                   (b) $150\sqrt{3}$ m                   (c) $150\sqrt{2}$ m                     (d) 75 m                                                                [CBSE 2014] Let AB be the tower and point C be the position of the car.

We have,

Hence, the correct answer is option (b).

#### Question 10:

Choose the correct answer of the following question:

A kite is flying at a height of 30 m from the ground. The length of string from the kite to the ground is 60 m. Assuming that there is no slack
in the string, the angle of elevation of the kite at the ground is

(a) 45°                   (b) 30°                   (c) 60°                   (d) 90°                                                                                                      [CBSE 2012] Let point A be the position of the kite and AC be its string.

We have,

Hence, the correct answer is option (b).

#### Question 11:

Choose the correct answer of the following question:

From the top of a cliff 20 m high, the angle of elevation of the top of a tower is found to be equal to the angle of depression of the foot of the
tower. The height of the tower is

(a) 20 m                    (b) 40 m                    (c) 60 m                    (d) 80 m                                                                                        [CBSE 2013C] Let AB be the cliff and CD be the tower.

We have,

Hence, the correct answer is option (b).

Disclaimer: The answer given in the textbook is incorrect. The same has been rectified above.

#### Question 12:

Choose the correct answer of the following question:

If a l.5-m-tall girl stands at a distance of 3 m from a lamp post and casts a shadow of length 4.5 m on the ground, then the height of the lamp post is

(a) 1.5 m                      (b) 2 m                      (c) 2.5 m                      (d) 2.8 m Let AB be the lamp post; CD be the girl and DE be her shadow.

We have,

Hence, the correct answer is option (c).

#### Question 13:

Choose the correct answer of the following question:

The length of the shadow of a tower standing on level ground is found to be 2x metres longer when the sun's elevation is 30° than when it was
45°. The height of the tower is

(a) $\left(2\sqrt{3}x\right)$ m                   (b) $\left(3\sqrt{2}x\right)$ m                    (c) $\left(\sqrt{3}-1\right)x$ m                    (d) $\left(\sqrt{3}+1\right)x$ m Let CD = h be the height of the tower.

We have,

Hence, the correct answer is option (d).

#### Question 14:

Choose the correct answer of the following question:

The lengths of a vertical rod and its shadow are in the ratio $1:\sqrt{3}$. The angle of elevation of the sun is

(a) 30°                    (b) 45°                     (c) 60°                     (d) 90° Let AB be the rod and BC be its shadow; and $\theta$ be the angle of elevation of the sun.

Hence, the correct answer is option (a).

#### Question 15:

Choose the correct answer of the following question:

A pole casts a shadow of length $2\sqrt{3}$ m on the ground when the sun's elevation is 60°. The height of the pole is

(a) $4\sqrt{3}$ m                    (b) 6 m                    (c) 12 m                    (d) 3 m                                                                     [CBSE 2015] Let AB be the pole and BC be its shadow.

We have,

Hence, the correct answer is option (b).

#### Question 16:

Choose the correct answer of the following question:

In the given figure, a tower AB is 20 m high and BC, its shadow on the ground is $20\sqrt{3}$ m long. The sun's altitude is

(a) 30°                   (b) 45°                   (c) 60°                   (d) none of these                                                                         [CBSE 2015] Let the sun's altitude be $\theta$.

We have,

Hence, the correct answer is option (a).

#### Question 17:

Choose the correct answer of the following question:

The tops of two towers of heights x and y, standing on a level ground subtend angles of 30° and 60°, respectively at the centre of the line joining their feet. Then, x : y is

(a) 1 : 2                  (b) 2 : 1                  (c) 1 : 3                  (d) 3 : 1                                                                                       [CBSE 2015] Let AB and CD be the two towers such that AB = and CD = y.

We have,

Hence, the correct answer is option (c).

#### Question 18:

The angle of elevation of the top of a tower from a point on the ground 30 m away from the foot of the tower is 30°. The height of the tower is

(a) 30 m
(b) $10\sqrt{3}\mathrm{m}$
(c) 20 m
(d) $10\sqrt{2}\mathrm{m}$

(b) $10\sqrt{3}\mathrm{m}$
Let $AB$ be the tower and $O$ be the point of observation.
Also,
and  m
Let:
m In ∆$AOB$, we have:

m

Hence, the height of the tower is $10\sqrt{3}$ m.

#### Question 19:

The string of a kite is 100 m long and it makes an angle of 60° with the horizontal. If these is no slack in the string, the height of the kite from the ground is

(a) $50\sqrt{3}\mathrm{m}$
(b) $100\sqrt{3}\mathrm{m}$
(c) $50\sqrt{2}\mathrm{m}$
(d) 100 m

(a) $50\sqrt{3}\mathrm{m}$
Let $AB$ be the string of the kite and $AX$ be the horizontal line.
If $BC$ ⊥ $AX$, then  m and ∠.
Let:
m In the right ∆$ACB$, we have:

m
Hence, the height of the kite is $50\sqrt{3}$ m.

#### Question 20:

If the angles of elevation of the top of a tower form tow points at distances a and b from the base and in the same straight line with it are complementary, then the height of the tower is

(a) $\sqrt{\frac{a}{b}}$
(b) $\sqrt{ab}$
(c) $\sqrt{a+b}$
(d) $\sqrt{a-b}$

(b) $\sqrt{ab}$
Let $AB$ be the tower and and $D$ be the points of observation on $AC$.
Let:
, ∠ and  m
Thus, we have:
and Now, in the right ∆ABC, we have:
⇒              ...(i)
In the right ∆ABD, we have:
⇒     ...(ii)

On multiplying (i) and (ii), we have:

[ ∵ ]
⇒
⇒  m

Hence, the height of the tower is $\sqrt{ab}$ m.

#### Question 21:

On the level ground, the angle of elevation of a tower is 30°. On moving 20 m nearer, the angle of elevation is 60°. The height of the tower is

(a) 10 m
(b) $10\sqrt{3}\mathrm{m}$
(c) 15 m
(d) $5\sqrt{3}\mathrm{m}$

(b) $10\sqrt{3}\mathrm{m}$
Let $AB$ be the tower and $C$ and $D$ be the points of observation such that ∠, ∠,  m  and  m. Now, in ∆$ADB$, we have:

⇒

In ∆$ACB$, we have:

⇒
∴
⇒
⇒  ⇒
∴ Height of the tower AB =  m

#### Question 22:

In a rectangle, the angle between a diagonal and a side is 30° and the length of this diagonal is 8 cm. the area of the rectangle is

(a) 16 cm2
(b) $\frac{16}{\sqrt{3}}{\mathrm{cm}}^{2}$
(c) $16\sqrt{3}{\mathrm{cm}}^{2}$
(d) $8\sqrt{3}{\mathrm{cm}}^{2}$

(c) $16\sqrt{3}{\mathrm{cm}}^{2}$
Let $ABCD$ be the rectangle in which ∠ and  cm. In ∆$BAC$, we have:

⇒  m
Again,

m
∴ Area of the rectangle =  cm2

#### Question 23:

From the top of a hill, the angles of depression of two consecutive km stones due east are found to be 30° and 45°. The height of the hill is

(a) $\frac{1}{2}\left(\sqrt{3}-1\right)\mathrm{km}$
(b) $\frac{1}{2}\left(\sqrt{3}+1\right)\mathrm{km}$
(c) $\left(\sqrt{3}-1\right)\mathrm{km}$
(d) $\left(\sqrt{3}+1\right)\mathrm{km}$

(b) $\frac{1}{2}\left(\sqrt{3}+1\right)\mathrm{km}$
Let $AB$ be the hill making angles of depression at points $C$ and $D$ such that ∠, ∠ and  km.
Let:
km and  km In ∆$ADB$, we have:

⇒   ⇒         ...(i)
In ∆$ACB$, we have:

...(ii)
On putting the value of $h$ taken from (i) in (ii), we get:

On multiplying the numerator and denominator of the above equation by $\left(\sqrt{3}+1\right)$, we get:
km

Hence, the height of the hill is $\frac{1}{2}\left(\sqrt{3}+1\right)$ km.

#### Question 24:

If the elevation of the sun changes form 30° to 60°, then the difference between the lengths of shadows of a pole 15 m high, is

(a) 7.5 m
(b) 15 m
(c) $10\sqrt{3}\mathrm{m}$
(d) $5\sqrt{3}\mathrm{m}$

(c) $10\sqrt{3}\mathrm{m}$
Let $AB$ be the pole and $AC$ and $AD$ be its shadows.
We have:
, ∠ and m In ∆$ACB$, we have:

⇒   ⇒  m

Now, in ∆$ADB$, we have:

⇒   ⇒    m

∴ Difference between the lengths of the shadows =  m

#### Question 25:

An observer 1.5 m tall 28.5 away from a tower and the angle of elevation of the top of the tower form the eye of the observer is 45°. The height of the tower is

(a) 27 m
(b) 30 m
(c) 28.5 m
(d) none of these

(b) 30 m
Let $AB$ be the observer and $CD$ be the tower. Draw $BE$ ⊥ $CD$, Let  metres. Then,
m ,  m and ∠.
=  m.
In right  ∆$BED$, we have:

⇒  m
Hence the height of the tower is $30$ m.

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