Find the best tutors and institutes for Class 10 Tuition
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.
Solution :
Given:
- Two poles of equal height, let this height be $h$ meters.
- The width of the road is $80$ m.
- A point $P$ is located on the road between the two poles.
- The angle of elevation from point $P$ to the top of the first pole is $60^\circ$.
- The angle of elevation from point $P$ to the top of the second pole is $30^\circ$.
To Find:
- The height of the poles ($h$).
- The distance of point $P$ from each pole.
Step 1: Define variables and set up the geometry.
Let the two poles be $AC$ and $BD$ such that $AC = BD = h$.
Let the width of the road $CD = 80$ m.
Let $P$ be a point on $CD$ such that $CP = x$ meters. Then, $PD = (80 - x)$ meters.
Step 2: Apply trigonometric ratios in the right-angled triangles.
In $\triangle ACP$, using the tangent ratio $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$:
$\tan(60^\circ) = \frac{AC}{CP}$
$\sqrt{3} = \frac{h}{x}$
$h = x\sqrt{3}$ --- (Equation 1)
In $\triangle BDP$:
$\tan(30^\circ) = \frac{BD}{PD}$
$\frac{1}{\sqrt{3}} = \frac{h}{80 - x}$
$h = \frac{80 - x}{\sqrt{3}}$ --- (Equation 2)
Step 3: Solve for $x$.
Equating Equation 1 and Equation 2:
$x\sqrt{3} = \frac{80 - x}{\sqrt{3}}$
Multiply both sides by $\sqrt{3}$:
$3x = 80 - x$
$3x + x = 80$
$4x = 80$
$x = 20$ m
Step 4: Calculate the height $h$ and the remaining distance.
Substitute $x = 20$ into Equation 1:
$h = 20\sqrt{3}$ m
The distance of point $P$ from the first pole is $CP = x = 20$ m.
The distance of point $P$ from the second pole is $PD = 80 - x = 80 - 20 = 60$ m.
Final Answer: The height of the poles is $20\sqrt{3}$ m (approx. $34.64$ m), and the distances of the point from the poles are $20$ m and $60$ m respectively.
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).
- 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.
- 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
I am Mopidevi Padmaja, an experienced Math Teacher, Quantitative Aptitude Coach, and Certified Vedic math trainer. I teach CBSE 9th and 10th Math, Quantitative Aptitude for Competitive exams (CAT, SSC, RRB, etc.) & Vedic Maths starting from grade 6 to graduation level students. -------------------------------------------------------------------------------------------------- I hold an Engineering degree & am an Associate Member of the Institution of Engineers (India) I am also a certified Vedic Math trainer -------------------------------------------------------------------------------------------------- I have over 10+ years of teaching experience both online & offline. I have taught over 1000+ students to date. I have been teaching online for the last 7 years for Indian and overseas students. I am familiar with various teaching platforms available online and comfortable using them. -------------------------------------------------------------------------------------------------- I have a passion for making Math the easiest, simple, and most fun subject for students. I work on various learning issues in Math, like Math phobia, focus problems, word problem understanding, concept understanding, confusion, time management in math exams, and silly mistakes which is a major problem for most students. I could help my students to score more than 90% marks in Maths subjects who were losing around 20 to 25 marks by confusion in concepts, silly mistakes, lack of time management in math exams, etc. I am successful in making Math an interesting and easy subject for my students who have hated Math. -------------------------------------------------------------------------------------------------- I thoroughly teach fundamentals, concepts, formulae, and guide with smart ways of doing calculations, smart ways of doing word problems, strategies for solving questions, etc. I use visual aids in teaching like 2D animated PowerPoint presentations. I make students comfortable and at ease and encourage them to ask their doubts during a session and even after sessions through messages. I respond to messages within a short time. If more students have doubts about a specific topic I even take a special session for them. I always try to improve my techniques of assessment by using a quiz, worksheets, test papers, Sample papers, etc., so that I can do 100 % assessment and take the necessary steps to improve my students. Every student is important to me. In each session, I teach according to the needs and understanding levels of my students. I work with students having various intellectual and behavioral capabilities. I learn to teach and teach to learn. My ultimate goal is to help the student to become an independent learner. I believe marks are not about grades and ranks but about building confidence levels in students. ---------------------------------------------------------------------------------------------- I continuously upgrade myself by attending workshops, and webinars, browsing the internet and reading lots of books. I use Digital Media for a lot of learning new developments in the field of my teaching. I attend webinars & seminars and interact with other teachers & professionals from across the world.
I would like to express my sincere gratitude to the teacher Padmaja Mopidevi for the exceptional guidance provided to my daughter throughout the CBSE 10th grade. Her dedication and teaching methods truly helped simplify complex mathematical concepts, making the subject much more approachable. Thanks to her constant support and encouragement, my daughter was able to secure a distinction in her board exams. We are incredibly grateful for the positive impact she has had on my daughter's academic journey and confidence. I highly recommend her tuition classes to any student looking to excel in mathematics.
I have more than 15 years of experience in teaching maths and science for class 10 .. I also prepare students for various competition exams like olympiads, NDA etc
She teaches better than every teacher she does conceptual clarity and gives more homework for practice and she helps me to give occasional tests to improve my score
I was using the urban pro for home tuitions queries. It was very useful and productive platform for the job seekers like me. It helps the needed in giving the contacts and mutually. I would like to avail further services.
I am an engineer and i have been teaching for the past 16 years. I put stress on the clarification of the basic concepts of the subject. I teach maths and science.
I am an experienced Spoken English and Class VI-X subject tutor with two years of experience in teaching. Currently, I am pursuing my bachelor's degree in BA from Magadh University. I can teach students who are in State, International Baccalaureate, CBSE and ICSE Board. My students have improved a lot through my teaching, and I've received five reviews till now with 100% positive feedback. I always try to understand my students and help them overcome their fear of failure in specific subjects like math and 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