###### Advertisements

###### Advertisements

A TV tower stands vertically on a bank of a canal. From a point on the 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 with of the canal.

###### Advertisements

#### Solution 1

In ΔABC,

`(AB)/(BC)` = tan 60º

`(AB)/(BC) = sqrt3`

`BC = (AB)/sqrt3`

In ΔABD,

`(AB)/(BD) `= tan 30º

`(AB)/(BC+CD) = 1/sqrt3`

`(AB)/((AB)/sqrt3+20) = 1/sqrt3`

`(ABsqrt3)/(AB+20sqrt3) = 1/sqrt3`

`3AB = AB+20sqrt3`

`2AB = 20sqrt3`

`AB = 10sqrt3 m`

`BC = (AB)/sqrt3 = ((10sqrt3)/sqrt3)m = 10 m`

Therefore, the height of the tower is `10sqrt3` m and the width of the canal is 10 m.

#### Solution 2

Let PQ=h m be the height of the TV tower and BQ= x m be the width of the canal.

We have,

AB = 20 m, ∠PAQ = 30° ,∠BQ = x and PQ = h

In ΔPBQ,

` tan 60° = (PQ)/(BQ)`

`⇒ sqrt(3) = h/x`

`⇒ h = x sqrt(3) ` ........................(1)

Again in ΔAPQ,

`tan 30° = (PQ)/(AQ)`

`⇒ 1/sqrt(3) = h/(AB +BQ)`

`⇒ 1/sqrt(3) = (x sqrt(3)) / (20+3)` [ Using (1)]

⇒ 3x = 20+x

⇒3x -x =20

⇒ 2x = 20

`⇒ x=20/2`

⇒ x = 10 m

Substituting x = 10 in (i),we get

`h = 10 sqrt(3)m`

So, the height of the TV tower is 10`sqrt( 3)` mand the width of the canal is 10 m.

#### RELATED QUESTIONS

A person standing on the bank of river observes that the angle of elevation of the top of a tree standing on the opposite bank is 60°. When he moves 40 m away from the bank, he finds the angle of elevation to be 30°. Find the height of the tree and width of the river. `(sqrt 3=1.73)`

There are three stair-steps as shown in the figure below. Each stair step has width 25 cm, height 12 cm and length 50 cm. How many bricks have been used in it, if each brick is 12.5 cm x 6.25 cm x 4 cm?

A 1.2 m tall girl spots a balloon moving with the wind in a horizontal line at a height of 88.2 m from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is 60°. After some time, the angle of elevation reduces to 30°. Find the distance travelled by the balloon during the interval.

The angles of depression of two ships from the top of a lighthouse and on the same side of it are found to be 45° and 30° respectively. If the ships are 200 m apart, find the height of the lighthouse.

An observer 1.5m tall is 30 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.

From the top of a lighthouse, an observer looking at a ship makes angle of depression of 60°. If the height of the lighthouse is 90 metre, then find how far the ship is from the lighthouse.

Two poles of heights 18 metre and 7 metre are erected on a ground. The length of the wire fastened at their tops in 22 metre. Find the angle made by the wire with the horizontal.

A kite is flying at a height of 60 m above the ground. The string attached to the kite is tied at the ground. It makes an angle of 60° with the ground. Assuming that the string is straight, find the length of the string.

A boy standing at a distance of 48 meters from a building observes the top of the building and makes an angle of elevation of 30°. Find the height of the building.

A ladder on the platform of a fire brigade van can be elevated at an angle of 70° to the maximum. The length of the ladder can be extended upto 20 m. If the platform is 2m above the ground, find the maximum height from the ground upto which the ladder can reach. (sin 70° = 0.94)

While landing at an airport, a pilot made an angle of depression of 20°. Average speed of the plane was 200 km/hr. The plane reached the ground after 54 seconds. Find the height at which the plane was when it started landing. (sin 20° = 0.342)

A storm broke a tree and the tree top rested on ground 20 m away from the

base of the tree, making an angle of 60o with the ground. Find the height

of the tree.

What is the angle of elevation of the Sun when the length of the shadow of a vetical pole is equal to its height?

*AB *is a pole of height 6 m standing at a point *B *and *CD *is a ladder inclined at angle of 60^{0 }to the horizontal and reaches upto a point D of pole . If *AD *= 2.54 m , find the length of the ladder.

The angle of elevation of the top of a tower standing on a horizontal plane from a point A is α. After walking a distance d towards the foot of the tower the angle of elevation is found to be β. The height of the tower is

If the angle of elevation of a cloud from a point 200 m above a lake is 30° and the angle of depression of its reflection in the lake is 60°, then the height of the cloud above the lake is

From the top of a lighthouse, an observer looks at a ship and finds the angle of depression to be 60° . If the height of the lighthouse is 84 meters, then find how far is that ship from the lighthouse? (√3 = 1.73)

If two circles having centers P and Q and radii 3 cm and 5 cm. touch each other externally, find the distance PQ.

Find the angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of a tower of height `10sqrt(3)` m

A road is flanked on either side by continuous rows of houses of height `4sqrt(3)` m with no space in between them. A pedestrian is standing on the median of the road facing a row house. The angle of elevation from the pedestrian to the top of the house is 30°. Find the width of the road

The top of a 15 m high tower makes an angle of elevation of 60° with the bottom of an electronic pole and angle of elevation of 30° with the top of the pole. What is the height of the electric pole?

From the top of a rock `50sqrt(3)` m high, the angle of depression of a car on the ground is observed to be 30°. Find the distance of the car from the rock

A lift in a building of height 90 feet with transparent glass walls is descending from the top of the building. At the top of the building, the angle of depression to a fountain in the garden is 60°. Two minutes later, the angle of depression reduces to 30°. If the fountain is `30sqrt(3)` feet from the entrance of the lift, find the speed of the lift which is descending.

The angle of elevation of the top of a cell phone tower from the foot of a high apartment is 60° and the angle of depression of the foot of the tower from the top of the apartment is 30°. If the height of the apartment is 50 m, find the height of the cell phone tower. According to radiation control norms, the minimum height of a cell phone tower should be 120 m. State if the height of the above mentioned cell phone tower meets the radiation norms

Three villagers A, B and C can see each other using telescope across a valley. The horizontal distance between A and B is 8 km and the horizontal distance between B and C is 12 km. The angle of depression of B from A is 20° and the angle of elevation of C from B is 30°. Calculate the vertical height between A and B. (tan 20° = 0.3640, `sqrt3` = 1.732)

Three villagers A, B and C can see each other using telescope across a valley. The horizontal distance between A and B is 8 km and the horizontal distance between B and C is 12 km. The angle of depression of B from A is 20° and the angle of elevation of C from B is 30°. Calculate the vertical height between B and C. (tan 20° = 0.3640, `sqrt3` = 1.732)

If the ratio of the height of a tower and the length of its shadow is `sqrt(3): 1`, then the angle of elevation of the sun has measure

The angle of elevation of a cloud from a point h metres above a lake is β. The angle of depression of its reflection in the lake is 45°. The height of location of the cloud from the lake is

The angle of elevation of the top of a tower from a certain point is 30°. If the observer moves 20 meters towards the tower, the angle of elevation of the top increases by 15°. Find the height of the tower ____________.

The angle of elevation of the top of a tower from two points distant s and t from its foot are complementary. Prove that the height of the tower is `sqrt(st)`

If one looks from a tower 10 m high at the top of a flag staff, the depression angle of 30° is made. Also, looking at the bottom of the staff from the tower, the angle of the depression made is of 60°. Find the height of the flag staff.

The top of a banquet hall has an angle of elevation of 45° from the foot of a transmission tower and the angle of elevation of the topmost point of the tower from the foot of the banquet hall is 60°. If the tower is 60 m high, find the height of the banquet hall in decimals.

The angles of elevation of the bottom and the top of a flag fixed at the top of a 25 m high building are 30° and 60° respectively from a point on the ground. Find the height of the flag.

A monkey is climbing a rope of length 15 m in a circus. The rope is tied to a vertical pole from its top. Find the height of the pole, if the angle, the rope makes with the ground level is equal to 60°.

**Read the following passage and answer the questions given below.**

Qutub Minar, located in South Delhi, India was built in the year 1193. It is 72 m high tower. Working on a school project, Charu and Daljeet visited the monument. They used trigonometry to find their distance from the tower. Observe the picture given below. Points C and D represent their positions on the ground in line with the base of tower, the angles of elevation of top of the tower (Point A) are 60° and 45° from points C and D respectively. |

- Based on the above information, draw a well-labelled diagram.
- Find the distances CD, BC and BD. [use `sqrt(3)` = 1.73]

From the base of a pole of height 20 meter, the angle of elevation of the top of a tower is 60°. The pole subtends an angle 30° at the top of tower. Then the height of tower is ______.

Two vertical poles are 150 m apart and the height of one is three times that of the other. If from the middle point of the line joining their feet, an observer finds the angles of elevation of their tops to be complementary, then the height of the shorter pole (in meters) is ______.

The angle of elevation of the top P of a vertical tower PQ of height 10 from a point A on the horizontal ground is 45°. Let R be a point on AQ and from a point B, vertically above R, the angle of elevation of P is 60°. If ∠BAQ = 30°, AB = d and the area of the trapezium PQRB is α, then the ordered pair (d, α) is ______.

Let AB and PQ be two vertical poles, 160 m apart from each other. Let C be the middle point of B and Q, which are feet of these two poles. Let `π/8` and θ be the angles of elevation from C to P and A, respectively. If the height of pole PQ is twice the height of pole AB, then, tan^{2} θ is equal to ______.

Two vertical poles AB = 15 m and CD = 10 m are standing apart on a horizontal ground with points A and C on the ground. If P is the point of intersection of BC and AD, then the height of P (in m) above the line AC is ______.

The angle of elevation of the top of a vertical tower from a point A, due east of it is 45°. The angle of elevation of the top of the same tower from a point B, due south of A is 30°. If the distance between A and B is `54sqrt(2)` m, then the height of the tower (in metres), is ______.

A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank of the river is 60° and when he retires 40 meters away from the tree the angle of elevation becomes 30°. The breadth of the river is ______.

The top of a hill when observed from the top and bottom of a building of height h is at angles of elevation p and q respectively. What is the height of the hill?

Two circles of radii 5 cm and 3 cm touch each other externally. Find the distance between their centres.