M L Aggarwal - Understanding ICSE Mathematics - Class 10 solutions for Chapter 19 Heights and Distances

A person standing on the bank of a river observes that the angle subtended by a tree on the opposite bank is 60⁰; when he retires 20 m from the bank, he finds the angle to be 30⁰. Find the height of the tree and the breadth of the river.

The upper part of a tree broken by wind falls to the ground without being detached. The top of the broken part touches the ground at an angle of 38⁰ 30’ at a point 6 m from the foot of the tree.

Calculate the height at which the tree is broken.

The upper part of a tree broken by wind falls to the ground without being detached. The top of the broken part touches the ground at an angle of 38⁰ 30’ at a point 6 m from the foot of the tree.

Calculate the original height of the tree correct to two decimal places.

From a point P on level ground, the angle of elevation of the top of a tower is 30⁰. If the tower is 100 m high, how far is P from the foot of the tower?

A ladder is placed against a wall such that it reaches the top of the wall. The foot of the ladder is 1.5 metres away from the wall and the ladder is inclined at an angle of 60⁰ with the ground. Find the height of the wall.

What is the angle of elevation of the sun when the length of the shadow of a vertical pole is equal to its height.

A river is 60 m wide. A tree of unknown height is on one bank. The angle of elevation of the top of the tree from the point exactly opposite to the foot of the tree on the other bank is 30⁰. Find the height of the tree.

In the adjoining figure, the angle of elevation from a point P of the top of a tower QR, 50 m high is 600 and that of the tower PT from a point Q is 300. Find the height of the tower PT, correct to the nearest metre.

From the top of a cliff 92 m high, the angle of depression of a buoy is 20⁰. Calculate to the nearest metre, the distance of the buoy from the foot of the cliff.

A boy is flying a kite with a string of length 100 m. If the string is tight and the angle of elevation of the kite is 26⁰ 32’, find the height of the kite correct to one decimal place, (ignore the height of the boy).

An electric pole is 10 m high. A steel wire tied to the top of the pole is affixed at a point on the ground to keep the pole upright. If the wire makes an angle of 45⁰ with the horizontal through the foot of the pole, find the length of the wire.

A bridge across a river makes an angle of 450 with the river bank. If the length of the bridge across the river is 200 metres, what is the breadth of the river.

A vertical tower is 20 m high. A man standing at some distance from the tower knows that the cosine of the angle of elevation of the top of the tower is 0.53. How far is he standing from the foot of the tower?

An observer 1.5 m tall is 20.5 metres away from a tower 22 metres high. Determine the angle of elevation of the top of the tower from the eye of the observer.

From a boat 300 metres away from a vertical cliff, the angles of elevation of the top and the foot of a vertical concrete pillar at the edge of the cliff are 550 40’ and 540 20’ respectively. Find the height of the pillar correct to the nearest metre.

The shadow of a vertical tower on a level ground increases by 10 m when the altitude of the sun changes from 45⁰ to 30⁰. Find the height of the tower, correct to two decimal places.

From a point P on the ground, the angle of elevation of the top of a 10 m tall building and a helicopter hovering over the top of the building are 30⁰ and 60⁰ respectively. Find the height of the helicopter above the ground.

An aeroplane when flying at a heigt of 3125 m from the ground passes vertically below another plane at an instant when the angles of elevation of the two planes from the same point on the ground are 30⁰ and 60⁰ respectively. Find the distance between the two planes at the instant.

From the top of a hill, the angles of depression of two consecutive kilometer stones, due east are found to be 30⁰ and 45⁰ respectively. Find the distance of two stones from the foot of the hill.

A man observes the angles of elevation of the top of a building to be 30⁰. He walks towards it in a horizontal line through its base. On covering 60 m the angle of elevation changes to 60⁰. Find the height of the building correct to the nearest decimal place.

An electric pole is 10 metres high. If its shadow is 10√3 metres in length, find the elevation of the sun.

The angles of depression of two ships A and B as observed from the top of a light house 60 m high are 60⁰ and 45⁰ respectively. If the two ships are on the opposite sides of the light house, find the distance between the two ships.

Give your answer correct to the nearest whole number.

An aeroplane at an altitude of 250 m observes the angle of depression of two boats on the opposite banks of a river to be 45⁰ and 60⁰ respectively.

Find the width of the river. Write the answer correct to the nearest whole number.

The angles of depression of the top and the bottom of an 8 m tall building from the top of a multi-storeyed building are 30⁰ and 45⁰ respectively.

Find the height of tire multi-storeyed building and the distance between the two buildings, correct to two decimal places.

From a tower 126 m high, the angles of depression of two rocks which are in a horizontal line through the base of the tower are 16⁰ and 12⁰ 20’.

Find the distance between the rocks if they are on the opposite sides of the tower.

A man 1.8 m high stands at a distance of 3.6 m from a lamp post and casts a shadow of 5.4 m on the ground.

Find the height of the lamp post.

An aircraft is flying at a constant height with a speed of 360 km/h. From a point on the ground, the angle of elevation of the aircraft at an instant was observed to be 45⁰. After 20 seconds, the angle of elevation was observed to be 30⁰.

Determine the height at which the aircraft flying. (use √3 = 1.732)

A pole of height 5 m is fixed on the top of a tower. The angle of elevation of the top of the pole as observed from a point A on the ground is 60⁰ and the angle of depression of the point A from the top of the tower is 45⁰.

Find the height of the tower. (Take √3 = 1.732)

A vertical pole and a vertical tower are on the same level ground. From the top of the pole the angle of elevation of the top of the tower is 60⁰ and the angle of depression of the foot of the tower is 30⁰.

Find the height of the tower if the height of the pole is 20 m.

At a point on level ground, the angle of elevation of a vertical lower is found to be such that its tangent is 5/12. On walking 192 m towards the tower, the tangent of the angle is found to be ¾.

Find the height of the tower.

The horizontal distance between two towers is 140 m. The angle of elevation of the top of the first tower when seen from the top of the second tower is 30⁰.

If the height of the second tower is 60 m, find the height of the first tower.

From the top of a building 20 m high, the angle of elevation of the top of a monument is 45⁰ and the angle of depression of its foot is 15⁰.

Find the height of the monument.

In the figure, not drawn to scale, TF is a tower. The elevation of T from A is x0 where tan x = 2/5 and AF = 200 m. The elevation of T from B, where AB = 80 m, is y0.

Calculate: the angle y, correct to the nearest degree.

From the top of a church spire 96 m high, the angles of depression of two vehicles on a road, at the same level as the base of the spire and on the same side of it are x⁰ and y⁰, where tan x⁰ = ¼ and tan y⁰ = 1/7.

Calculate the distance between the vehicles.

In the adjoining figure, not drawn to the scale, AB is a tower and two objects C and D are located on the ground, on the same side of AB. When observed from the top A of the tower, their angles of depression are 450 and 600.

Find the distance between the two objects. If the height of the tower is 300. Give your answer to the nearest metre.

In the figure, not drawn to scale, TF is a tower. The elevation of T from A is x0 where tan x = 2/5 and AF = 200 m. The elevation of T from B, where AB = 80 m, is y0.

Calculate: the height of the tower TF.

The angle of elevation of a pillar from a point A on the ground is 45⁰ and from a point B diametrically opposite to A and on the other side of the pillar is 60⁰.

Find the height of the pillar, given that the distance between A and B is 15 m.

As observed from the top of a 80 m tall light house, the angles of depression of two ships on the same side of the light house in horizontal line with its base are 30⁰ and 40⁰ respectively.

Find the distance between the two ships. Give your answer correct to the nearest metre.

From a tower 126 m high, the angles of depression of two rocks which are in a horizontal line through the base of the tower are 16⁰ and 12⁰ 20’.

Find the distance between the rocks if they are on the same side of the tower.

From two points A and B on the same side of a building, the angles of elevation of the top of the building are 30⁰ and 60⁰ respectively.

If the height of the building is 10 m, find the distance between A and B correct to two decimal places.

The angle of elevation of the top of an unfinished tower at a point distant 120 m from its base is 45⁰.

How much higher must the tower be raised so that its angle of elevation at the same point may be 60⁰?

In the adjoining figure, the shadow of a vertical tower on the level ground increases by 10 m, when the altitude of the sun changes from 450 to 300.

Find the height of the tower and give your answer, correct to 1/10 of a metre.

The angle of elevation of the top of a tower from a point on the ground and at a distance of 150 m from its foot is 30⁰. Find the height of the tower correct to one place of decimal.

M l aggarwal understanding icse mathematics class 10 Solutions for Chapter 19 heights and distances