NCERT Solutions For Class 10 Math Chapter – 9 Exercise – 9.1

NCERT Solutions For Class 10 Chapter – 9 Some Applications of Trigonometry Exercise – 9.1

NCERT Solutions For Class 10 Chapter – 9 Some Applications of Trigonometry provides comprehensive solutions for all the questions in the NCERT Textbooks To excel in the second term examinations, NCERT Solutions will increase the level of confidence among the students, as the concepts are clearly explained and structure The solutions are prepared and reviewed by our subjects matter experts and they are revised according to the latest CBSE Syllabus for 2021 – 2022 and guidelines of the CBSE Board.

It covers all the chapters and provides chapter – wise solutions. These NCERT Solutions for Class 10 Maths are helpful for the students to clarify their doubts and provide a strong foundation for every concept. With the help of NCERT Solutions for Class 10 every student should be capable of solving the complex problem in each exercise

Q1. 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 30⁰

$C:\Users\User\Desktop\NCERT\images\tri-prob1.JPG$

Solution:-

In right ∆ABC , using sine formula

Sin 30⁰ = AB/AC

1/2 = AB/20

AB = 10 m

Therefore , the height of the pole is 10 m.

Q2. A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30⁰ with it. The distances between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree.

Solution:-

In right ∆ABC ,

Cos 30⁰= BC/AC

√3/2 = 8/AC

AC = 16/√3.

$C:\Users\User\Desktop\NCERT\images\trigprob2.jpg$

Also ,

Tan 30⁰ = AB/BC

1/√3 = AB/8

AB = 8/√3.

Therefore , total height of the tree = AB + AC = 16/√3 + 8/√3 = 24/√3 = 8√3.

Q3. A contractor plans to install two slide 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 30⁰ to the ground, whereas for elder children , she wants to have a sleep slide at a height of 3 m , and inclined at an angel of 60⁰ to the ground What should be the length of the slide in each case ?

Solution:-

Ncert solutions class 10 chapter 9-3

$C:\Users\User\Desktop\NCERT\images\trig3.jpg$

In right ∆ABC ,

Sin 30⁰ = AB/AC

1/2 = 1.5/AC

AC = 3

In right ∆PQR ,

Sin 60⁰ = PQ/PR

√3/2 = 3/PR

PR = 2√3.

Length of the slide for below 5 = 3m and

Length of the slide for elder children = 2√3 m.

Q4. The angle of elevation of the top of a tower from a point on the ground, which id 30 m away from the foot of the tower , is 30⁰ find the height of the tower.

Solution:-

$C:\Users\User\Desktop\NCERT\images\tri4.jpg$

In right ABC

Tan 30⁰ = AB/AC

AB = 10√3.

Thus , the height of the tower is 10√3 m.

Q5. 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 60⁰ Find the length of the string , assuming that there is no slack in the string.

Solution:-

$C:\Users\User\Desktop\NCERT\images\trig5.jpg$

In ∆CAB

Sin 60⁰ = BC/AC

√3/2 = 60/AC

AC = 40√3 m.

The length of the string from the ground is 40√3 m.

Q6. A 1.5 m tall boy is standing at some distances from a 30 m tall building. The angle of elevation from his eyes to the top of the building increases from 30⁰to 60⁰ as he walks towards the building Find the distance he walked towards the building.

Solution:-

$C:\Users\User\Desktop\NCERT\images\trig6.jpg$

In right ∆ABD ,

Tan 30⁰ = AB/BD

1/√3 = 28.5/BD

BD = 28.5 √3 m.

In right ∆ABC ,

Tan 60⁰ = AB/BC

√3 = 28.5/ BC

BC = 28.5√3/3

Therefore , the length of BC is 28.5√3/3 m.

XY = CD = BD – BC = (28.5√3 – 28.5√3/3)

= 19√3 m.

Thus , the distance boy walked towards the building is 19√3 m.

Q7. 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 20 m high building are 45⁰and 60⁰ respectively. Find the height of the tower.

Solution:-

In right ∆BCD ,

Tan 45⁰= BC/CD

1 = 20/CD

CD = 20

In right ∆ACD,

Tan 60⁰ = AC/AD

√3 = AC/20

AC = 20√3

$C:\Users\User\Desktop\NCERT\images\trig7.jpg$

AB = AC – BC = (20√3 – 20)

= 20(√3 – 1) m.

Q8. A statue 1.6 m tall , stand on the top of a pedestal. From a point on the ground , the angle of elevation of the top of the statue is 60⁰ and from the same point the angle of elevation of the top of the pedestal is 45⁰ Find the height of the pedestal.

Solution:-

$C:\Users\User\Desktop\NCERT\images\trig8.jpg$

In right triangle BCD ,

Tan 45⁰ = BC/CD

1 = BC/CD

BC = CD (I)

In right ∆ACD,

Tan 60⁰ = AC/AD

√3 = (AB + BC)/CD

√3 CD = 1.6 + BC

√3BC = 1.6 + BC

√3BC – BC = 1.6

BC(√3 – 1) = 1.6

BC = 1.6/ (√3 – 1) m

BC = 0.8 (√3 + 1)

Thus , the height of the pedestal is 0.8 ( √3 + 1) m.

Q9. The angle elevation of the top of a building from the foot of the tower is 30⁰ and the angle of elevation of the top of the tower from the foot of the building is 60⁰ If the tower is 50 m high , find the height of the building.

Solution:-

In right ∆BCD,

Tan 60⁰ = CD/BC

√3 = 50/BC

BC = 50/√3.

In right ∆ABC ,

Tan 30⁰ = AB/AC

1/√3 = AB/BC

AB = 50/3

Thus , height of the building is 50/3 m.

Q10. 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 60⁰ and 30⁰ respectively . Find the height of the poles and the distances of the point from the poles.

Solution:-

Let AB and CD be the poles of equal height.

O is the point between them from where the height of elevation taken. BD is the distance between the poles.

As per above figure, AB = CD,

OB + OD = 80 m

Now,

In right ΔCDO,

tan 30° = CD/OD

1/√3 = CD/OD

CD = OD/√3 … (1)

Again,

In right ΔABO,

tan 60° = AB/OB

√3 = AB/(80-OD)

AB = √3(80-OD)

AB = CD (Given)

√3(80-OD) = OD/√3 (Using equation (1))

3(80-OD) = OD

240 – 3 OD = OD

4 OD = 240

OD = 60

Putting the value of OD in equation (1)

CD = OD/√3

CD = 60/√3

CD = 20√3 m

Also,

OB + OD = 80 m

⇒ OB = (80-60) m = 20 m

Thus, the height of the poles are 20√3 m and distance from the point of elevation are 20 m and

60 m respectively.

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 joing this point to the foot of the tower, the angle of elevation of the top of the tower is 30° (see Fig. 9.12). Find the height of the tower and the width of the canal.

Ncert solutions class 10 chapter 9-12

Solution: Given, AB is the height of the tower.

DC = 20 m (given)

As per given diagram, In right ΔABD,

tan 30° = AB/BD

1/√3 = AB/(20+BC)

AB = (20+BC)/√3 … (i)

Again,

In right ΔABC,

tan 60° = AB/BC

√3 = AB/BC

AB = √3 BC … (ii)

From equation (i) and (ii)

√3 BC = (20+BC)/√3

3 BC = 20 + BC

2 BC = 20

BC = 10

Putting the value of BC in equation (ii)

AB = 10√3

This implies, the height of the tower is 10√3 m and the width of the canal is 10 m.

From the top of a 7 m 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.

Solution:

Let AB be the building of height 7 m and EC be the height of the tower.

A is the point from where elevation of tower is 60° and the angle of depression of its foot is 45°.

EC = DE + CD

Also, CD = AB = 7 m. and BC = AD

To Find: EC = Height of the tower

Design a figure based on given instructions:

In right ΔABC,

tan 45° = AB/BC

1= 7/BC

BC = 7

Since BC = AD

So AD = 7

Again, from right triangle ADE,

tan 60° = DE/AD

√3 = DE/7

⇒ DE = 7√3 m

Now: EC = DE + CD

= (7√3 + 7) = 7(√3+1)

Therefore, Height of the tower is 7(√3+1) m. Answer!

As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.

Solution:

Let AB be the lighthouse of height 75 m. Let C and D be the positions of the ships.

30° and 45° are the angles of depression from the lighthouse.

Draw a figure based on given instructions:

To Find: CD = distance between two ships

Step 1: From right triangle ABC,

tan 45° = AB/BC

1= 75/BC

BC = 75 m

Step 2: Form right triangle ABD,

tan 30° = AB/BD

1/√3 = 75/BD

BD = 75√3

Step 3: To find measure of CD, use results obtained in step 1 and step 2.

CD = BD – BC = (75√3 – 75) = 75(√3-1)

The distance between the two ships is 75(√3-1) m. Answer!

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° (see Fig. 9.13). Find the distance travelled by the balloon during the interval.

Solution:

Let the initial position of the balloon be A and final position be B.

Height of balloon above the girl height = 88.2 m – 1.2 m = 87 m.

To Find: Distance travelled by the balloon = DE = CE – CD

Let us redesign the given figure as per our convenient

Step 1: In right ΔBEC,

tan 30° = BE/CE

1/√3= 87/CE

CE = 87√3

Step 2:

In right ΔADC,

tan 60° = AD/CD

√3= 87/CD

CD = 87/√3 = 29√3

Step 3:

DE = CE – CD = (87√3 – 29√3) = 29√3(3 – 1) = 58√3

Distance travelled by the balloon = 58√3 m.

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 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 from this point.

Solution:

Let AB be the tower.

D is the initial and C is the final position of the car respectively.

Since man is standing at the top of the tower so, Angles of depression are measured from A.

BC is the distance from the foot of the tower to the car.

Step 1: In right ΔABC,

tan 60° = AB/BC

√3 = AB/BC

BC = AB/√3

AB = √3 BC

Step 2:

In right ΔABD,

tan 30° = AB/BD

1/√3 = AB/BD

AB = BD/√3

Step 3: Form step 1 and Step 2, we have

√3 BC = BD/√3 (Since LHS are same, so RHS are also same)

3 BC = BD

3 BC = BC + CD

2BC = CD

or BC = CD/2

Here, distance of BC is half of CD. Thus, the time taken is also half.

Time taken by car to travel distance CD = 6 sec. Time taken by car to travel BC = 6/2 = 3 sec.

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.

Solution:

Let AB be the tower. C and D be the two points with distance 4 m and 9 m from the base respectively. As per question,