Find the best tutors and institutes for Class 10 Tuition
Q12:
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 :
Given:
1. Height of the building ($AB$) = $7$ m.
2. Angle of elevation of the top of the tower ($EC$) from the top of the building ($A$) = $60^\circ$.
3. Angle of depression of the foot of the tower ($D$) from the top of the building ($A$) = $45^\circ$.
To find:
The total height of the cable tower ($CD$).
Step 1: Define the variables and geometric relationships.
Let $AB$ be the building of height $7$ m. Let $CD$ be the cable tower. Let $A$ be the top of the building and $B$ be its foot. Let $C$ be the top of the tower and $D$ be its foot. Let $E$ be a point on $CD$ such that $AE$ is horizontal. Thus, $AE = BD$ and $ED = AB = 7$ m.
Step 2: Analyze triangle $\triangle ABD$.
In the right-angled triangle $\triangle ABD$, the angle of depression from $A$ to $D$ is $45^\circ$. Therefore, the angle of elevation from $D$ to $A$ is also $45^\circ$ (alternate interior angles).
Using the trigonometric ratio tangent:
$\tan(45^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AB}{BD}$
Since $\tan(45^\circ) = 1$:
$1 = \frac{7}{BD}$
$BD = 7$ m
[Since $AE$ is parallel to $BD$, $AE = BD = 7$ m].
Step 3: Analyze triangle $\triangle AEC$.
In the right-angled triangle $\triangle AEC$, the angle of elevation is $60^\circ$.
$\tan(60^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{CE}{AE}$
We know $\tan(60^\circ) = \sqrt{3}$ and $AE = 7$ m:
$\sqrt{3} = \frac{CE}{7}$
$CE = 7\sqrt{3}$ m
Step 4: Calculate the total height of the tower.
The total height of the tower $CD = CE + ED$.
We know $ED = AB = 7$ m.
$CD = 7\sqrt{3} + 7$
$CD = 7(\sqrt{3} + 1)$ m
Using the approximation $\sqrt{3} \approx 1.732$:
$CD \approx 7(1.732 + 1) = 7(2.732) = 19.124$ m.
Final Answer: The height of the tower is $7(\sqrt{3} + 1)$ m or approximately $19.12$ m.
More Questions from Class 10 Mathematics Applications of Trigonometry EXERCISE 9.1
- 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 $30°$ (see Fig. 9.11).
- 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 $60°$ and $30°$, respectively. Find the height of the poles and the distances of the point from the poles.
- Q11: 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.
- Q13: 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.
- Q14: 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.
- Q15: 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.
- 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 $30°$ 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.
- Q3: 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 $30°$ 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 $60°$ to the ground. What should be the length of the slide in each case?
- Q4: 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.
- 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 $60°$. Find the length of the string, assuming that there is no slack in the string.
- Q6: 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.
- 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 a $20$ m high building are $45°$ and $60°$ respectively. Find the height of the tower.
- Q8: 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.
- Q9: 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.
CBSE Solutions for Class 10 Mathematics Applications of Trigonometry
Chapters in CBSE - Class 10 Mathematics
Top Tutors who teach Applications of Trigonometry
Top Tutor
Chopasni Housing Board, Jodhpur
5 yrs of
Exp
Online Classes
Tutor's home
My aim is to prepare students in there best ability and help them achieve good marks. My students scored above 80% in all subjects in class 10th
Top Tutor
Sector-63 Mohali, Chandigarh
5 yrs of
Exp
1 student
11 hrs of classes completed
500
per hour
Online Classes
Tutor's home
I have been tutoring students from past 6 years and as a professional School teacher from past 4 years . I know what is the competition and how to make student competent to face that with effective learning material and concept clarity .
Excellent experience. Service was wonderful. There was ambience habitat. All the things was understand easily.
Top Tutor
Sector 83, Gurgaon
20 yrs of
Exp
4 students
3838 hrs of classes completed
650
per hour
Online Classes
Tutor's home
For me, teaching is part of learning. As I am teaching I am also learning from my learners as they are diverse. Each learner in the classroom brings his or her unique background and ability. These are the elements that I invest in my teaching. Moreover, I allow learners to be themselves. This is a way of creating a conducive environment for learning.
Her method of teaching is very clear and student friendly. It’s really very easy for me to understand and also she has a lot of patience.
Top Tutor
Burul Rd Burul, Burul
5 yrs of
Exp
100
per hour
Online Classes
Student's Home
Tutor's home
I am a teacher of math. I am giving home/online/tutor home tuition since 2016. My key skills are computer language. I am b. Tech graduate. I can teach short tricks that can help students in their future life.
Top Tutor
Malkajgiri D.D.Colony, Hyderabad
20 yrs of
Exp
2 students
1316 hrs of classes completed
400
per hour
Online Classes
I was teaching in Government and Government aided schools. I Providing supervision & consultation for students, in the form of academic advising, students’ support, recruitment, courses and programs. Also, participating in student & staff social and cultural activities. PPP Approach to make the students understand the topics. Unified Approach to make the students to work independently and collaboratively. Heuristic Approach to make the students to understand theories through lab practical. Use few other contemporary approaches not limited to Brain Storming, Mind Mapping, Concept Mapping, Snowballing etc.,
Find more Tutor for Applications of Trigonometry in your City
- Bangalore Mathematics Tutors
- Delhi Mathematics Tutors
- Chennai Mathematics Tutors
- Gurgaon Mathematics Tutors
- Noida Mathematics Tutors
- Hyderabad Mathematics Tutors
- Mumbai Mathematics Tutors
- Ghaziabad Mathematics Tutors
- Chandigarh Mathematics Tutors
- Pune Mathematics Tutors
- Jaipur Mathematics Tutors
- Surat Mathematics Tutors
Download free CBSE - Class 10 Mathematics Applications of Trigonometry EXERCISE 9.1 worksheets
Download Now
