NCERT Solutions for Class 10 Maths Exercise 9.1 Chapter 9 Some Applications of Trigonometry – FREE PDF Download

NCERT Class 10 Maths Ch 9 is one of the most important ones in the NCERT syllabus. Duly following NCERT Solutions for Class 10 Maths
Chapter 9 ensures you that there will be no hindrance when you opt for more advanced branches of Maths. This is where CoolGyan comes in. Our free Class 10 Some Applications of Trigonometry solutions will help you understand the chapter thoroughly.

NCERT Solutions for Class 10 Maths Chapter 9 – Some Applications of Trigonometry

1. A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is (see figure).

Ans. In right triangle ABC,

AB= 20/2

AB = 10 m

Hence, the height of the pole is 10 m.

2. A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree.

Ans. Let AC be the broken part of tree

In right triangle ABC,

Again,

AB = m

Height of the tree = AB + AD=AB+AC

= =

= = m

3. A contractor plans to install two slides for the children to play in a park. For the children below the age of 5 years, she prefers to have a slide whose top is at a height of 1.5 m and is inclined at an angle of to the ground, whereas for elder children, she wants to have a steep slide at a height of 3 m and inclined at an angle of to the ground. What should be the length of the slide in each case?

Ans. In right triangle ABC,

AC = 3 m

In right triangle PQR,

PR = m

Hence, the lengths of the slides are 3 m and m respectively.

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

Ans. In right triangle ABC, AB be the height of the tower.

AB = m

= m

5. A kite is flying at a height of 60 m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is Find the length of the string, assuming that there is no slack in the string.

Ans. In right triangle ABC, AC is the length of the string

AC = m

Hence the length of the string is m.

6. A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of elevation from his eyes to the top of the building increases from to as he walks towards the building. Find the distance he walked towards the building.

Ans. AB = 30 m and PR = 1.5 m

AC = AB – BC

= AB – PR (As, BC = PR)

= 30 – 1.5

= 28.5 m

In right triangle ACQ,

QC = m

In right triangle ACP,

PQ =

PQ = = m

Hence, the distance the boy walked towards the building is m.

7. From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a 20 m high building are and respectively. Find the height of the tower.

Ans. Let the height of the tower be

m. Then, in right triangle CBP,

……….(i)

In right triangle ABP,

BP = 20 m

Putting this value in eq. (i), we get,

The height of the tower is m.

8. A statue, 1.6 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 and from the same point the angle of elevation of the top of the pedestal is Find the height of the pedestal.

Ans. Let the height of the pedestal be

BC = m

In right triangle ACP,

……….(i)

In right triangle BCP,

PC =

[From eq. (i)]

Hence, the height of the pedestal is m.

9. The angle of elevation of the top of a building from the foot of the tower is and the angle of elevation of the top of the tower from the foot of the building is If the tower is 50 m high, find the height of the building.

Ans. Let the height of the building be

In right triangle PQB,

BQ = m……….(i)

In right triangle ABQ,

BQ = m……….(ii)

From eq. (i) and (ii),

10. Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are and respectively. Find the height of the poles and the distances of the point from the poles.

Ans. Let the height of each poles be H m

AB = PQ = H

In right triangle PRQ,

(in figure change AB= H m rather than h m)

H = m……….(i)

In right triangle ABR,

[From eq. (i)]

H = = m

Also, BR = = 80 – 20 = 60 m

Hence the heights of the poles are m each and the distances of the point from poles are 20 m and 60 m respectively.

11. 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 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 (see figure). Find the height of the tower and the width of the canal.

Ans. Let AB be the TV tower .

In right triangle ABC,

AB = m……….(i)

In right triangle ABD,

AB = m……….(ii)

From eq. (i) and (ii),

3BC = BC + 20

BC = 10 m

From eq. (i), AB = m

Hence height of the tower is m and the width of the canal is 10 m.

12. From the top of a 7 m high building, the angle of elevation of the top of a cable tower is and the angle of depression of its foot is Determine the height of the tower.

Ans. In right triangle ABD,

BD = 7 m

AE = 7 m

In right triangle AEC,

CE = m

CD = CE + ED

= CE + AB (As AB = ED)

= = m

Hence height of the tower is m.

13. As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are and If one ship is exactly behind the other on the same side of the lighthouse, find the distance between two ships.

Ans. In right triangle ABQ,

BQ = 75 m……….(i)

In right triangle ABP,

[From eq. (i)]

75 + QP =

QP = m

Hence the distance between the two ships is m.

14. 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 distant is 60⁰ After some time, the angle of elevation reduces to 30⁰ (see figure). Find the distance travelled by the balloon during the interval.

Ans. As, per question;

AB = PQ = 88.2 – 1.2 = 87 m

In right triangle ABC,

$\sqrt{3} = \frac{87}{B C}$

$B C = \frac{87}{\sqrt{3}} = 29 \sqrt{3}$ m

In right triangle PQC,

$\frac{1}{\sqrt{3}} = \frac{87}{29 \sqrt{3} + B Q}$

$29 \sqrt{3} + B Q = 87 \sqrt{3}$

$B Q = 58 \sqrt{3}$ m

Hence the distance travelled by the balloon during the interval is $58 \sqrt{3}$ m.

15. A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of , 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 Find the time taken by the car to reach the foot of the tower from this point.

Ans. In right triangle ABP,

BP = AB ……….(i)

In right triangle ABQ,

BQ = ……….(ii)

PQ = BP – BQ

PQ = AB

= = = 2BQ [From eq. (ii)]

BQ = PQ

Time taken by the car to travel a distance PQ = 6 seconds.

Time taken by the car to travel a distance BQ, i.e. PQ = x 6 = 3 seconds.

Hence, the further time taken by the car to reach the foot of the tower is 3 seconds.

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

Ans. Let