Find the best tutors and institutes for Class 10 Tuition
Q8:
From a solid cylinder whose height is $2.4$ cm and diameter $1.4$ cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid to the nearest cm$^2$.
Solution :
Given:
- Height of the cylinder ($h$) = $2.4$ cm
- Diameter of the cylinder ($d$) = $1.4$ cm
- Radius of the cylinder ($r$) = $\frac{d}{2} = \frac{1.4}{2} = 0.7$ cm
- A conical cavity of the same height ($h = 2.4$ cm) and same radius ($r = 0.7$ cm) is hollowed out.
To Find:
The total surface area of the remaining solid, rounded to the nearest cm$^2$.
Step 1: Identify the components of the Total Surface Area (TSA)
When a conical cavity is hollowed out from a solid cylinder, the total surface area of the remaining solid consists of:
- The curved surface area of the cylinder ($2\pi rh$)
- The area of the circular base of the cylinder ($\pi r^2$)
- The curved surface area of the conical cavity ($\pi rl$)
Formula: $TSA = 2\pi rh + \pi r^2 + \pi rl$
Step 2: Calculate the slant height ($l$) of the cone
The slant height $l$ is given by the Pythagorean theorem: $l = \sqrt{r^2 + h^2}$
$l = \sqrt{(0.7)^2 + (2.4)^2}$
$l = \sqrt{0.49 + 5.76}$
$l = \sqrt{6.25} = 2.5$ cm
Step 3: Calculate the individual areas
Using $\pi \approx \frac{22}{7}$:
Curved Surface Area of Cylinder = $2 \times \frac{22}{7} \times 0.7 \times 2.4 = 2 \times 22 \times 0.1 \times 2.4 = 10.56$ cm$^2$
Area of circular base = $\pi r^2 = \frac{22}{7} \times 0.7 \times 0.7 = 22 \times 0.1 \times 0.7 = 1.54$ cm$^2$
Curved Surface Area of Cone = $\pi rl = \frac{22}{7} \times 0.7 \times 2.5 = 22 \times 0.1 \times 2.5 = 5.5$ cm$^2$
Step 4: Sum the areas
$TSA = 10.56 + 1.54 + 5.5$
$TSA = 17.6$ cm$^2$
Step 5: Rounding to the nearest cm$^2$
Since $17.6$ is closer to $18$ than $17$, we round to $18$ cm$^2$.
Final Answer: 18 cm$^2$
More Questions from Class 10 Mathematics Surface Areas and Volumes EXERCISE 12.1
- Q1: 2 cubes each of volume $64$ cm$^3$ are joined end to end. Find the surface area of the resulting cuboid.
- Q2: A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is $14$ cm and the total height of the vessel is $13$ cm. Find the inner surface area of the vessel.
- Q3: A toy is in the form of a cone of radius $3.5$ cm mounted on a hemisphere of same radius. The total height of the toy is $15.5$ cm. Find the total surface area of the toy.
- Q4: A cubical block of side $7$ cm is surmounted by a hemisphere. What is the greatest diameter the hemisphere can have? Find the surface area of the solid.
- Q5: A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter $l$ of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid.
- Q6: A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends (see Fig. 12.10). The length of the entire capsule is $14$ mm and the diameter of the capsule is $5$ mm. Find its surface area.
- Q7: A tent is in the shape of a cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are $2.1$ m and $4$ m respectively, and the slant height of the top is $2.8$ m, find the area of the canvas used for making the tent. Also, find the cost of the canvas of the tent at the rate of $₹ 500$ per m$^2$. (Note that the base of the tent will not be covered with canvas.)
- Q9: A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in Fig. 12.11. If the height of the cylinder is $10$ cm, and its base is of radius $3.5$ cm, find the total surface area of the article.
CBSE Solutions for Class 10 Mathematics Surface Areas and Volumes
Chapters in CBSE - Class 10 Mathematics
Top Tutors who teach Surface Areas and Volumes
I am an IIM Kashipur alumnus, passionate about guiding students to achieve their immediate and long-term goals. As a personal tutor and founder of the online educational startup ‘MinDely,’ I have gained deep insights into child psychology and student needs, allowing me to tailor my teaching methods accordingly. I design lesson plans based on each student's grasp of concepts, ensuring a student-friendly and easy-to-understand approach. My weekly tests provide valuable insights for parents to track their child’s progress. I take pride in the success of my students, many of whom have graduated from IITs, DTU, and NSIT.
I have 5years of teaching experience where I implement Inductive ,Deductive teaching techinques in mathematics. I begin the teaching concepts of maths with basic level to high level which ensures the teaching learning process occurs even a below average student I make special work sheets for the brain nutrient students and also to the below average students ,so that helps students to become learn individually according to their level. In teaching maths concepts regarding Geometry , Trignometry and Areas and Circles here I use the concrete to semi concrete from thier i take them to abstract so child can easily familiar to the concept, teaching learning process occurs in a positive way. In teaching maths I use heuristic method which creates an enthusiasm among the students to learn maths subject and it eradicate the fear of maths from the students. As a teacher I teach according to the students taste and their interests, so teaching learning process will be occurred in a meaningful way. I observe the individual difference of the students while teaching, so that I can plan to teach according individual difference of the different students in the classroom. I use remedial class for the slow learners to develop in the subject. In this remedial session I mostly concentrating on the basic of maths subject so children can overcome the previous misunderstanding concepts regarding maths and here it will be taught his doubts regarding subject will be clarified. By teaching maths subject I make students all round development which is very valuable in the life.
With my extensive background in mathematical studies, as well as my teaching certification in science and mathematics, it is my pleasure to post this information for your review for an experienced Math Teacher. My experience and enthusiasm for educating youth provide me with the drive and knowledge to significantly impact your school in this role. Lesson planning needs assessment, and visually engaging educational techniques are just a few areas in which I have gained experience and thorough comprehension. Throughout my academic career, I honed my communication, creativity, and motivational skills. My commitment to fostering collaborative and exciting educational environments positions me to successfully encourage and develop productive and accomplished students. With over 10 years of hands-on successful teaching experience, I am confident in my ability and passion to become a positive addition to this UrbanPro community of experienced educators.
Jaya mam helped me to learn different concepts of mathematics. She is very experienced and helpful. Thanks.
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.
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.
Its been a really nice teaching experience with Mr.Amit so far. He is very knowledgeable and a patient teacher. He always clarifies the concepts clearly and ensures good understanding.
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 12.1 worksheets
Download Now