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## Exercise 9.1 Page No: 203

**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 30° (see fig. 9.11).**

**Solution:**

The length of the rope is 20 m, and the angle made by the rope with the ground level is 30°.

**Given: ** AC = 20 m and angle C = 30°

**To find**: Height of the pole

Let AB be the vertical pole

In the right ΔABC, using the sine formula

sin 30° = AB/AC

^{Using the value of sin 30 degrees is ½, we have}

^{ 1/2 = AB/20}

^{ AB = 20/2}

^{AB = 10}

Therefore, the height of the pole is 10 m.

**2. A tree breaks due to a storm, and the broken part bends so that the top of the tree touches the ground making an angle 30° 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.**

**Solution: **

Using the given instructions, draw a figure. Let AC be the broken part of the tree. Angle C = 30°

BC = 8 m

To find: Height of the tree, which is AB

From figure: Total height of the tree is the sum of AB and AC, i.e., AB+AC

In the right ΔABC,

Using Cosine and tangent angles,

cos 30° = BC/AC

We know that, cos 30° = √3/2

√3/2 = 8/AC

AC = 16/√3 …(1)

Also,

tan 30° = AB/BC

1/√3 = AB/8

AB = 8/√3 ….(2)

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

**3. A contractor plans to install two slides for the children to play in a park. For 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 30° to the ground, whereas for elder children, she wants to have a steep slide at a height of 3m, and inclined at an angle of 60° to the ground. What should be the length of the slide in each case?**

**Solution**:

As per the contractor’s plan,

Let ABC is the slide inclined at 30° with length AC, and PQR be the slide inclined at

60° with length PR.

To Find: AC and PR

In the right ΔABC,

sin 30° = AB/AC

1/2 = 1.5/AC

AC = 3

Also,

In the right ΔPQR,

sin 60° = PQ/PR

⇒ √3/2 = 3/PR

⇒ PR = 2√3

Hence, the length of the slide below 5 = 3 m and

Length of the slide for elders children = 2√3 m

**4. 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 the tower, is 30°. Find the height of the tower.**

**Solution:**

Let AB be the height of the tower and C be the point elevation, which is 30 m away from the foot of the tower.

To find: AB (height of the tower)

In the right ABC

tan 30° = AB/BC

1/√3 = AB/30

⇒ AB = 10√3

Thus, the height of the tower is 10√3 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 60°. Find the length of the string, assuming that there is no slack in the string.**

**Solution:**

Draw a figure based on the given instructions,

Let BC = Height of the kite from the ground, BC = 60 m

AC = Inclined length of the string from the ground and

A is the point where the string of the kite is tied.

To find: Length of the string from the ground, i.e., the value of AC

From the above figure,

sin 60° = BC/AC

⇒ √3/2 = 60/AC

⇒ AC = 40√3 m

Thus, the length of the string from the ground is 40√3 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 30° to 60° as he walks towards the building. Find the distance he walked towards the building.**

**Solution:**

Let the boy initially stand at point Y with an inclination 30°, and then he approaches the building to point X with an inclination 60°.

To find: The distance boy walked towards the building, i.e., XY

From figure,

XY = CD.

Height of the building = AZ = 30 m

AB = AZ – BZ = 30 – 1.5 = 28.5

The measure of AB is 28.5 m

In the right ΔABD,

tan 30° = AB/BD

1/√3 = 28.5/BD

BD = 28.5√3 m

Again,

In the right ΔABC,

tan 60° = AB/BC

√3 = 28.5/BC

BC = 28.5/√3 = 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) = 28.5√3(1-1/3) = 28.5√3 × 2/3 = 57/√3 m.

Thus, the distance boy walked towards the building is 57/√3 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 45° and 60°, respectively. Find the height of the tower. **

**Solution:**

Let BC be the 20 m high building.

D is the point on the ground from where the elevation is taken.

Height of transmission tower = AB = AC – BC

To find: AB, the height of the tower

From the figure, in the right ΔBCD,

tan 45° = BC/CD

1 = 20/CD

CD = 20

Again,

In the right ΔACD,

tan 60° = AC/CD

√3 = AC/20

AC = 20√3

Now, AB = AC – BC = (20√3-20) = 20(√3-1)

Height of transmission tower = 20(√3 – 1) 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 60°, and from the same point, the angle of elevation of the top of the pedestal is 45°. Find the height of the pedestal.**

**Solution:**

Let AB be the height of the statue.

D is the point on the ground from where the elevation is taken.

To find: The height of pedestal = BC = AC-AB

From figure,

In the right triangle BCD,

tan 45° = BC/CD

^{ 1 = BC/CD}

^{ BC = CD …..(1)}

Again,

In the right ΔACD,

tan 60° = AC/CD

^{ √3 = ( AB+BC)/CD }

^{ √3CD = 1.6 + BC}

^{ √3BC = 1.6 + BC (using equation (1)}

^{ √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.

**9. The angle of elevation of the top of a building from the foot of the tower is 30°, and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 50 m high, find the height of the building.**

**Solution:**

Let CD be the height of the tower. AB be the height of the building. BC be the distance between the foot of the building and the tower. The elevation is 30 degrees and 60 degrees from the tower and the building, respectively.

In the right ΔBCD,

tan 60° = CD/BC

^{ √3 = 50/BC}

^{ BC = 50/√3 …(1)}

Again,

In the right ΔABC,

tan 30° = AB/BC

^{⇒ 1/√3 = AB/BC }

Use the result obtained in equation (1)

^{AB = 50/3}

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

**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 60° and 30°, 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 is taken. BD is the distance between the poles.

As per the above figure, AB = CD,

OB + OD = 80 m

Now,

In the right ΔCDO,

tan 30° = CD/OD

1/√3 = CD/OD

CD = OD/√3 … (1)

Again,

In the 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 is 20√3 m, and the distance from the point of elevation is 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 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° (see Fig. 9.12). Find the height of the tower and the width of the canal.**

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

DC = 20 m (given)

As per the given diagram, in the right ΔABD,

tan 30° = AB/BD

1/√3 = AB/(20+BC)

AB = (20+BC)/√3 … (i)

Again,

In the right ΔABC,

tan 60° = AB/BC

√3 = AB/BC

AB = √3 BC … (ii)

From equations (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 that the height of the tower is 10√3 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 60°, and the angle of depression of its foot is 45°. Determine the height of the tower.**

**Solution**:

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

A is the point from where the elevation of the 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 the given instructions.

In the right ΔABC,

tan 45° = AB/BC

1= 7/BC

BC = 7

Since BC = AD

So AD = 7

Again, from the 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, the height of the tower is 7(√3+1) m.

**13. As observed from the top of a 75 m high lighthouse from the sea level, the angles of depression of the 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 the given instructions.

To find: CD = distance between two ships

Step 1: From the 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 the measure of CD, use the 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.

**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 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 the final position be B.

Height of the balloon above the girl’s 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 at our convenience.

Step 1: In the right ΔBEC,

tan 30° = BE/CE

1/√3= 87/CE

CE = 87√3

Step 2:

In the 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.

**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 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 position, and C is the final position of the car.

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

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

Step 1: In the right ΔABC,

tan 60° = AB/BC

√3 = AB/BC

BC = AB/√3

AB = √3 BC

Step 2:

In the 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 L.H.S. is the same, so R.H.S. is also the same)

3 BC = BD

3 BC = BC + CD

2BC = CD

or BC = CD/2

Here, the 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.

**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.**

**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

In the right ΔABC,

tan x = AB/BC

tan x = AB/4

AB = 4 tan x … (i)

Again, from right ΔABD,

tan (90°-x) = AB/BD

cot x = AB/9

AB = 9 cot x … (ii)

Multiplying equations (i) and (ii)

AB^{2} = 9 cot x × 4 tan x

⇒ AB^{2} = 36 (because cot x = 1/tan x

⇒ AB = ± 6

Height cannot be negative. Therefore, the height of the tower is 6 m.

Hence proved.

In Class 10, Exercise 9.1 recollects the concept of the trigonometric function to measure the height and distance of the objects without measuring them. By using sine, cosine and tangent functions, the actual measurement can be done by introducing some important terms like

- Line of sight
- Angle of Elevation
- Line of Depression
- Horizontal line

With the help of these terms and trigonometric functions, the height and distance of an object can be easily measured. Once students learn to find the angle of elevation and depression for a problem, they will easily find which trigonometric functions should be used to find the solution to a problem. Learn the entire NCERT Solutions for Class 10 Maths Chapter 9 to score more in board exams by understanding the applications of trigonometry in real-life situations.

Very nice question