Class 11 Exam > Class 11 Questions > If the curved surface of right circular cylin...
Start Learning for Free
If the curved surface of right circular cylinder inscribed in a sphere of radius r is maximum show the height of the cylinder is√2r?
Most Upvoted Answer
If the curved surface of right circular cylinder inscribed in a sphere...
Explanation:
We are given a right circular cylinder inscribed in a sphere of radius r. We need to find the height of the cylinder such that the curved surface area of the cylinder is maximum.
Let's consider a cross-section of the cylinder and the sphere that contains it.
Step 1: Deriving the Formula for Curved Surface Area of Cylinder
The curved surface area of the cylinder is given by:
C.S.A. = 2πrh
where r is the radius of the cylinder and h is the height of the cylinder.
Step 2: Deriving the Formula for Radius of Cylinder
The radius of the cylinder can be found using the Pythagorean theorem as follows:
r^2 + h^2 = (2r)^2
Simplifying the above equation, we get:
h^2 = 3r^2
Therefore, the radius of the cylinder is:
r = h/√3
Step 3: Deriving the Formula for Curved Surface Area of Cylinder in terms of h
Substituting the value of r in the formula for C.S.A., we get:
C.S.A. = 2πrh = 2π(h/√3)h = (2π/√3)h^2
Step 4: Finding the Maximum Value of C.S.A.
To find the maximum value of the curved surface area, we differentiate the formula for C.S.A. with respect to h and equate it to zero.
d/dh [(2π/√3)h^2] = (4π/√3)h = 0
Solving the above equation, we get:
h = √2r
Therefore, the height of the cylinder that maximizes the curved surface area is √2r.
Conclusion:
Hence, we have shown that the height of the cylinder is √2r when the curved surface area of the cylinder is maximum.
We are given a right circular cylinder inscribed in a sphere of radius r. We need to find the height of the cylinder such that the curved surface area of the cylinder is maximum.
Let's consider a cross-section of the cylinder and the sphere that contains it.
Step 1: Deriving the Formula for Curved Surface Area of Cylinder
The curved surface area of the cylinder is given by:
C.S.A. = 2πrh
where r is the radius of the cylinder and h is the height of the cylinder.
Step 2: Deriving the Formula for Radius of Cylinder
The radius of the cylinder can be found using the Pythagorean theorem as follows:
r^2 + h^2 = (2r)^2
Simplifying the above equation, we get:
h^2 = 3r^2
Therefore, the radius of the cylinder is:
r = h/√3
Step 3: Deriving the Formula for Curved Surface Area of Cylinder in terms of h
Substituting the value of r in the formula for C.S.A., we get:
C.S.A. = 2πrh = 2π(h/√3)h = (2π/√3)h^2
Step 4: Finding the Maximum Value of C.S.A.
To find the maximum value of the curved surface area, we differentiate the formula for C.S.A. with respect to h and equate it to zero.
d/dh [(2π/√3)h^2] = (4π/√3)h = 0
Solving the above equation, we get:
h = √2r
Therefore, the height of the cylinder that maximizes the curved surface area is √2r.
Conclusion:
Hence, we have shown that the height of the cylinder is √2r when the curved surface area of the cylinder is maximum.
Attention Class 11 Students!
To make sure you are not studying endlessly, EduRev has designed Class 11 study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Class 11.
|
Explore Courses for Class 11 exam
|
|
Top Courses for Class 11View all
If the curved surface of right circular cylinder inscribed in a sphere of radius r is maximum show the height of the cylinder is√2r?
Question Description
If the curved surface of right circular cylinder inscribed in a sphere of radius r is maximum show the height of the cylinder is√2r? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about If the curved surface of right circular cylinder inscribed in a sphere of radius r is maximum show the height of the cylinder is√2r? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the curved surface of right circular cylinder inscribed in a sphere of radius r is maximum show the height of the cylinder is√2r?.
If the curved surface of right circular cylinder inscribed in a sphere of radius r is maximum show the height of the cylinder is√2r? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about If the curved surface of right circular cylinder inscribed in a sphere of radius r is maximum show the height of the cylinder is√2r? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the curved surface of right circular cylinder inscribed in a sphere of radius r is maximum show the height of the cylinder is√2r?.
Solutions for If the curved surface of right circular cylinder inscribed in a sphere of radius r is maximum show the height of the cylinder is√2r? in English & in Hindi are available as part of our courses for Class 11.
Download more important topics, notes, lectures and mock test series for Class 11 Exam by signing up for free.
Here you can find the meaning of If the curved surface of right circular cylinder inscribed in a sphere of radius r is maximum show the height of the cylinder is√2r? defined & explained in the simplest way possible. Besides giving the explanation of
If the curved surface of right circular cylinder inscribed in a sphere of radius r is maximum show the height of the cylinder is√2r?, a detailed solution for If the curved surface of right circular cylinder inscribed in a sphere of radius r is maximum show the height of the cylinder is√2r? has been provided alongside types of If the curved surface of right circular cylinder inscribed in a sphere of radius r is maximum show the height of the cylinder is√2r? theory, EduRev gives you an
ample number of questions to practice If the curved surface of right circular cylinder inscribed in a sphere of radius r is maximum show the height of the cylinder is√2r? tests, examples and also practice Class 11 tests.
|
|
Explore Courses for Class 11 exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup