Find the best tutors and institutes for Class 10 Tuition
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.
Solution :
Given:
- Height of the building ($BC$) = $20$ m.
- The transmission tower ($CD$) is fixed on top of the building.
- Angle of elevation from a point $A$ on the ground to the bottom of the tower (point $C$) = $45^\circ$.
- Angle of elevation from point $A$ on the ground to the top of the tower (point $D$) = $60^\circ$.
To find:
The height of the transmission tower ($h = CD$).
Step 1: Define variables and identify triangles.
Let $AB = x$ be the distance from the point on the ground to the base of the building. Let $CD = h$ be the height of the tower. The total height of the building and tower is $BD = BC + CD = 20 + h$.
Step 2: Analyze $\triangle ABC$ (Right-angled at $B$).
In $\triangle ABC$, the angle of elevation $\angle BAC = 45^\circ$.
Using the trigonometric ratio: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$
$\tan(45^\circ) = \frac{BC}{AB}$
Since $\tan(45^\circ) = 1$ [Trigonometric table value]:
$1 = \frac{20}{x}$
$x = 20$ m [Equation 1]
Step 3: Analyze $\triangle ABD$ (Right-angled at $B$).
In $\triangle ABD$, the angle of elevation $\angle BAD = 60^\circ$.
$\tan(60^\circ) = \frac{BD}{AB}$
Since $\tan(60^\circ) = \sqrt{3}$ [Trigonometric table value]:
$\sqrt{3} = \frac{20 + h}{x}$
Step 4: Solve for $h$.
Substitute $x = 20$ from Equation 1 into the equation from Step 3:
$\sqrt{3} = \frac{20 + h}{20}$
$20\sqrt{3} = 20 + h$ [Multiplying both sides by 20]
$h = 20\sqrt{3} - 20$
$h = 20(\sqrt{3} - 1)$
Using $\sqrt{3} \approx 1.732$:
$h = 20(1.732 - 1)$
$h = 20(0.732)$
$h = 14.64$ m
Final Answer: The height of the transmission tower is $20(\sqrt{3} - 1)$ m or approximately $14.64$ 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.
- 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.
- 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.
- 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
Bajaj Nagar, Nagpur
3 yrs of
Exp
1 student
Online Classes
Student's Home
Coaching is about improvising a student from where he/she is at. Not about converting an already bright student into an achiever. It should be obvious. The whole idea of ONE ON ONE coaching is to improvise and to elimate the problems in a detailed manner, exactly the way a student desires. To connect and respond to a student cannot be everybodys cup of tea. This is where a capable one comes in to fill the gaps where modern day educationalists fail to fill.
Top Tutor
SCB Nagar, Kothagudem
20 yrs of
Exp
31 students
1966 hrs of classes completed
450
per hour
Online Classes
22 Years Experience in Schools IN ICSE/CBSE/IGCSE IN DIFFERENT STATE BOARDS
Sir is very friendly teacher he will support us more he is the best teacher of the year his teaching skills are really good.
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 12, Noida
6 yrs of
Exp
4 students
148 hrs of classes completed
400
per hour
Online Classes
I have worked as PGT teacher in one of the best CBSE schools and all of my students gained more than 90 percent marks. I have full concept of subjects and can assist students in gaining more than 90 percent marks.
She’s a excellent teacher , her behaviour her talking style is very sweet and if you didn’t understand any topic she’ll repeat it for you as many times you want!! Satisfied and loved her teaching
Top Tutor
Munirka, Delhi
3 yrs of
Exp
1 student
48 hrs of classes completed
200
per hour
Online Classes
I am a teacher since last 3-4 years. I have teaching tuition to students of class 9th and 10th CBSE board. In these covid times i prefer online classes and my subjects of expertise are science, maths ans social science.
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
- Chandigarh Mathematics Tutors
- Pune Mathematics Tutors
- Ghaziabad Mathematics Tutors
- Jaipur Mathematics Tutors
- Surat Mathematics Tutors
Download free CBSE - Class 10 Mathematics Applications of Trigonometry EXERCISE 9.1 worksheets
Download Now
