**GSEB Solutions for Class 10 mathematics – Heights and Distances (English Medium)**

### Exercise-10

**Question 1:**

A pole stands vertically on the ground. If the angle of elevation of the top of the pole from a point 90 m away from the pole has measure 30, find the height of the pole.

**Solution :**

**Question 2:**

A string of a kite is 100 m long and it makes an angle of measure 60 with the horizontal. Find the height of the kite, assuming that there is no slack in the string.

**Solution :**

**Question 3:**

A circus artist is climbing from the ground along a rope stretched from the top of a vertical pole and tied at the ground. The height of pole is 10 m and the angle made by the rope with ground level has measure 30. Calculate the distance covered by the artist in climbing to the top of the pole.

**Solution :**

**Question 4:**

A tree breaks due to a storm and the broken part bends such that the top of the tree touches the ground making an angle having measure 30 with the ground. The distance from the foot of the tree to the point where the top touches the ground is 30 m. Find the height of the tree.

**Solution :**

**Question 5:**

An electrician has to repair an electric fault on the pole of height 5 m. He needs to reach a point 2 m 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 measure 60 to the horizontal would enable him to reach the required position.

**Solution :**

**Question 6:**

As observed from a fixed point on the bank of a river, the angle of elevation of a temple on the opposite bank has measure 30. If the height of the temple is 20 m, find the width of the river.

**Solution :**

**Question 7:**

As observed from the top of a hill 200 m high, the angles of depression of two vehicles situated on the same side of the hill are found to have measure 30 and 60 respectively. Find the distance between the two vehicles.

**Solution :**

**Question 8:**

A person standing on the bank of a river, observes that the angle subtended by a tree on the opposite bank has measure 60. When he retreats 20 m from the bank, he finds the angle to have measure 30. Find the height of the tree and the breadth of the river.

**Solution :**

**Question 9:**

The shadow of a tower is 27 m, when the angle of elevation of the sun has measure 30. When the angle of elevation of the sun has measure 60, find the length of the shadow of the tower.

**Solution :**

**Question 10:**

From a point at the height 100 m above the sea level, the angles of depression of a ship in the sea is found to have measure 30. After some time the angle of depression of the ship has measure 45. Find the distance travelled by the ship during that time interval.

**Solution :**

**Question 11:**

From the top of a 300 m high light-house, the angles of depression of the top and foot of a tower have measure 30 and 60. Find the height of the tower.

**Solution :**

**Question 12:**

As observed from a point 60 m above a lake, the angle of elevation of an advertising ballon has measure 30 and from the same point the angle of depression of the image of the ballon in the lake has measure 60. Calculate the height of the balloon above the lake.

**Solution :**

**Question 13:**

Watching from a window 40 m high of a multistoreyed building, the angle of elevation of the top of a tower is found to have measure 45. The angle of elevation of the top of the same tower from the bottom of the building is found to have measure 60. Find the height of the tower.

**Solution :**

**Question 14:**

Two pillars of equal height stand on either side of a road, which is 100 m wide. The angles of elevation of the top of the pillars have measure 60 and 30 at a point on the road between the pillars. Find the position of the point from the nearest end of a pillars and the height of pillars.

**Solution :**

**Question 15:**

The angles of elevation of the top of a tower from two points at distance a and b metres from the base and in the same straight line with it are complementary. Prove that the height of the tower is \(\sqrt{ab}\) metres.

**Solution :**

**Question 16:**

A man on the top of a vertical tower observes a car moving at a uniform speed coming directly towards it. If it takes 12 minutes for the angle of depression to change its measure from 30 to 45, how soon after this, will the car reach the tower ?

**Solution :**

**Question 17:**

If the angle of elevation of a cloud from a point h metres above a lake has measure and the angle of depression of its reflection in the lake has measure 13, prove that the height of the could is \(\frac{h(\tan \beta +\tan \alpha )}{\tan \beta -\tan \alpha }m\)

**Solution :**

**Question 18:**

From the top of a building \(\overline{AB}\), 60 m high, the angles of depression of the top and bottom at a vertical lamp post \(\overline{CD}\) are observed to have measure 30 and 60 respectively. Find,

- the horizontal distance between building and lamp post.
- the height of the lamp post.
- the difference between the heights of the building and the lamp post

**Solution :**

**Question 19:**

A bridge across a valley is h metres long. There is a temple in the valley directly below the bridge. The angles of depression of the top of the temple from the two ends of the bridge have measures OC and B. Prove that the height of the bridge above the top of the temple is \(\frac{h(\tan \alpha \cdot \tan \beta )}{\tan \alpha +\tan \beta }\).

**Solution :**

**Question 20:**

At a point on level ground, the angle of elevation of a vertical tower is found to be such that its tangent is \(\frac{5}{12}\) On walking 192 metres towards the tower, the tangent of the angle is found to be \(\frac{3}{4}\). Find the height of the tower.

**Solution :**

**Question 21:**

A statue 1.46 m tall, stands on the top of a pedestal. From the point on the ground the angle of elevation of the top of the statue has measure 60 and from the same point, the angle of elevation of the top of the pedestal has measure 45. Find the height of the pedestal.

**Solution :**

**Question 22:**

Select a proper option (a), (b), (c) or (d) from given options :

**Question 22(1):**

On walking ………… metres on a hill making an angle of measure 30 with the ground, one can reach the height of ‘a’ metres from the ground.

**Solution :**

**Question 22(2):**

The angle of elevation of the top of the tower from a point P on the ground has measure 45. The distance of the tower from the point P is a and height of the tower is b. Then, ……….

**Solution :**

**Question 22(3):**

A 3 m long ladder leans on the wall such that its lower end remains 1.5 m away from the base of the wall. Then, the ladder makes an angle of measure ………. with the ground.

**Solution :**

**Question 22(4):**

A tower is 50\(\sqrt{3}\) m high. The angle of elevation of its top from a point 50 m away from its foot has measure ……..

**Solution :**

**Question 22(5):**

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

**Solution :**

**Question 22(6):**

If the angles of elevation of a tower from two points distance a and b (a > b) from its foot on the same side of the tower have measure 30 and 60, then the height of the tower is ……….

**Solution :**

**Question 22(7):**

The tops of two poles of height 18 m and 12 m are connected by a wire. If the wire makes an angle of measure 30 with horizontal, then the length of the wire is ……..

**Solution :**

**Question 22(8):**

The angle of elevation of the top of the building A from the base of building B has measure 50. The angle of elevation of the top of the building B from the base of building

- A has measure 70. Then,……….
- building A is taller than building B.
- Building B is taller than building A.
- Building A and building B are equally tall.
- The relation about the heights of A and B cannot be determined.

**Solution :**

**Question 22(9):**

If the angle of elevation of the top of a tower of a distance 400 m from its foot has measure 30, then the height of the tower is ……..

**Solution :**

**Question 22(10):**

The angle of depression of a ship from the top of a tower 30 m height has measure 60. Then, the distance of the ship from the base of the tower is ……….

**Solution :**

**Question 22(11):**

When the length of the shadow of the pole is equal to the height of the pole, then the angle of elevation source of light has measure ………..

**Solution :**

**Question 22(12):**

From the top of a building h metre high, the angle of depression of an object on the ground has measure O. The distance (in metres) of the object from the foot of the building is……..

**Solution :**

**Question 22(13):**

As observed from the top of the light house the angle of depression of the two ships P and Q anchored in the sea to the same side are found to have measure 35 and 50 respectively. Then from the light house….

**Solution :**

c. the distance of P is more than Q.

From the figure, we can see that,

The distance of P from the lighthouse is more than the distance of Q from the lighthouse.

**Question 22(14):**

Two poles are x metres apart and the height of one is double than that of the other. If from the mid-point of the line joining their feet, an observer finds the angle of elevation of their tops to be complementary, then the height of the shorter pole is ……

**Solution :**