Find the best tutors and institutes for Class 10 Tuition
Q5:
A vessel is in the form of an inverted cone. Its height is $8$ cm and the radius of its top, which is open, is $5$ cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius $0.5$ cm are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.
Solution :
Given:
1. A vessel in the shape of an inverted cone with height $h_c = 8$ cm and radius $r_c = 5$ cm.
2. The vessel is filled with water to the brim.
3. Lead shots are spherical in shape with radius $r_s = 0.5$ cm.
4. When lead shots are dropped, $\frac{1}{4}$ of the water in the cone flows out.
To Find:
The number of lead shots ($n$) dropped into the vessel.
Step 1: Calculate the volume of the conical vessel.
The formula for the volume of a cone is $V_c = \frac{1}{3}\pi r_c^2 h_c$.
$V_c = \frac{1}{3} \times \pi \times (5)^2 \times 8$
$V_c = \frac{1}{3} \times \pi \times 25 \times 8$
$V_c = \frac{200}{3}\pi \text{ cm}^3$
Step 2: Determine the volume of water that flows out.
According to the problem, the volume of water that flows out is equal to $\frac{1}{4}$ of the total volume of the cone.
$V_{out} = \frac{1}{4} \times V_c$
$V_{out} = \frac{1}{4} \times \frac{200}{3}\pi = \frac{50}{3}\pi \text{ cm}^3$
Step 3: Calculate the volume of one spherical lead shot.
The formula for the volume of a sphere is $V_s = \frac{4}{3}\pi r_s^3$.
$V_s = \frac{4}{3} \times \pi \times (0.5)^3$
$V_s = \frac{4}{3} \times \pi \times 0.125$
$V_s = \frac{4}{3} \times \pi \times \frac{1}{8} = \frac{1}{6}\pi \text{ cm}^3$
Step 4: Formulate the equation to find the number of lead shots ($n$).
The volume of water displaced is equal to the total volume of the $n$ lead shots dropped into the vessel [By Archimedes' Principle].
$n \times V_s = V_{out}$
$n \times (\frac{1}{6}\pi) = \frac{50}{3}\pi$
Step 5: Solve for $n$.
Divide both sides by $\pi$:
$n \times \frac{1}{6} = \frac{50}{3}$
Multiply both sides by 6:
$n = \frac{50}{3} \times 6$
$n = 50 \times 2$
$n = 100$
Final Answer: The number of lead shots dropped in the vessel is 100.
More Questions from Class 10 Mathematics Surface Areas and Volumes EXERCISE 12.2
- Q1: A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to $1$ cm and the height of the cone is equal to its radius. Find the volume of the solid in terms of $\pi$.
- Q2: Rachel, an engineering student, was asked to make a model shaped like a cylinder with two cones attached at its two ends by using a thin aluminium sheet. The diameter of the model is $3$ cm and its length is $12$ cm. If each cone has a height of $2$ cm, find the volume of air contained in the model that Rachel made. (Assume the outer and inner dimensions of the model to be nearly the same.)
- Q3: A gulab jamun, contains sugar syrup up to about $30\%$ of its volume. Find approximately how much syrup would be found in $45$ gulab jamuns, each shaped like a cylinder with two hemispherical ends with length $5$ cm and diameter $2.8$ cm (see Fig. 12.15).
- Q4: A pen stand made of wood is in the shape of a cuboid with four conical depressions to hold pens. The dimensions of the cuboid are $15$ cm by $10$ cm by $3.5$ cm. The radius of each of the depressions is $0.5$ cm and the depth is $1.4$ cm. Find the volume of wood in the entire stand (see Fig. 12.16).
- Q6: A solid iron pole consists of a cylinder of height $220$ cm and base diameter $24$ cm, which is surmounted by another cylinder of height $60$ cm and radius $8$ cm. Find the mass of the pole, given that $1$ cm$^3$ of iron has approximately $8$g mass. (Use $\pi = 3.14$)
- Q7: A solid consisting of a right circular cone of height $120$ cm and radius $60$ cm standing on a hemisphere of radius $60$ cm is placed upright in a right circular cylinder full of water such that it touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is $60$ cm and its height is $180$ cm.
- Q8: A spherical glass vessel has a cylindrical neck $8$ cm long, $2$ cm in diameter; the diameter of the spherical part is $8.5$ cm. By measuring the amount of water it holds, a child finds its volume to be $345$ cm$^3$. Check whether she is correct, taking the above as the inside measurements, and $\pi = 3.14$.
CBSE Solutions for Class 10 Mathematics Surface Areas and Volumes
Chapters in CBSE - Class 10 Mathematics
Top Tutors who teach Surface Areas and Volumes
Top Tutor
Sardar Patel Nagar, Dhanbad
6 yrs of
Exp
60 hrs of classes completed
Online Classes
I am an experienced, qualified teacher and tutor with over 5 years of experience in teaching Maths and phy , across different boards and diffrent entrance exam including IIT-JEE , Neet ,B.tech. Passionate about solving mathematical problems and physics problem, over the years I have helped thousands of students overcome their fear of Maths and physics.
Super Tutor
Katira Maharaja Hata, Arrah
4 yrs of
Exp
24 students
1791 hrs of classes completed
700
per hour
Online Classes
Tutor's home
I've been teaching Class 10th students of schools from India and abroad. I primarily work with them to prepare for Olympiad, JEE foundation and school exams. I've been teaching students of class 8th-10th Science and Mathematics for more than 3 years. So far, I have taught more than 100 students from India and abroad. I teach international students Mathematics, Physics and Chemistry. I teach almost all boards of India such as CBSE, ICSE and state boards. I also teach IGCSE and IB students. I've my offline coaching institute where I am teaching students of CBSE and state boards who require help in Science and Mathematics. Fee displayed on my profile is for 1×1 classes. For group classes, the fee is substantially reduced.
Top Tutor
Gharoli Gharoli, Delhi
16 yrs of
Exp
1 student
23 hrs of classes completed
2000
per hour
Online Classes
I bring with me 17 years of diverse experience in the pharmaceutical industry along with academic expertise, combining strong practical knowledge with a passion for teaching. I am dedicated to simplifying complex concepts and delivering them in an engaging manner, ensuring that students not only gain subject knowledge but also develop the skills, confidence, and potential required to achieve academic excellence and become industry-ready professionals.
I highly recommend this teacher to anyone looking for a clear and effective learning experience. The teaching style is simple, engaging, and easy to follow, which makes even difficult topics understandable. Every chapter is explained in depth, with attention to important concepts and details, ensuring that students build a strong foundation. The teacher is patient, knowledgeable, and dedicated to helping students succeed. Lessons are well-structured, and examples are used effectively to make learning more practical and enjoyable. This approach boosts confidence, improves understanding, and encourages active participation. Overall, an excellent teacher who makes learning both meaningful and enjoyable.
Top Tutor
Patel Nagar, Gurgaon
13 yrs of
Exp
300
per hour
Online Classes
Student's Home
Tutor's home
Top Tutor
Mahabubnagar, Mahabubnagar
9 yrs of
Exp
2 students
524 hrs of classes completed
400
per hour
Online Classes
Tutor's home
I am a B.Tech graduate. I completed my graduation in 2007. Because of passion for Teaching I entered in Teaching field. I have been teaching since 2014 in school/institute and at my home. I can teach mathematics and physical science subjects for 8, 9, 10 classes. I can make the students to understand the subject easily by taking relevant examples. This is my skill. I am passionate and dedicated.
I recently used UrbanPro to find a tutor for my child and was thoroughly impressed. The website is user-friendly and easy to navigate. I was able to search for tutors based on location, subject, and experience level. The profiles of the tutors were detailed, providing information about their qualifications, teaching style, and reviews from previous students. This helped me make an informed decision. Once I found a suitable tutor, the booking process was straightforward. The tutor was punctual and professional. They tailored their lessons to my child's needs and made learning enjoyable. Overall, I would highly recommend UrbanPro to anyone looking for quality services in areas like tutoring, beauty, home services, and more.
Find more Tutor for Surface Areas and Volumes 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 Surface Areas and Volumes EXERCISE 12.2 worksheets
Download Now
