Maxima & Minima: Inscribing Cylinder in Sphere of Radius R

In summary, to find the maximum surface area of a cylinder inscribed in a sphere of radius R, begin by establishing the relationship between the radius and length of the cylinder. This can be done by drawing a picture of the sphere and cylinder and setting up a coordinate system. Once the equation for the cylinder's surface area as a function of one variable is determined, solve for the maximum value.
  • #1
rishiraj20006
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How would we find the maximum surface area of a cylinder inscribed in a sphere of radius R. This problem is given in my textbook . I know concept of maxima and minima will be apllicable here but i can,t start and make the expression of surface area in a suitable manner. Anybody having answers
 
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  • #2
Start by determining what will be the relation (equation) between the radius of the cylinder and it's lenght, given that it is inscribed in the sphere. With that you can write the equation of the area of the cylinder as a function of 1 variable only (radius or lenght), and solve for the max.
 
  • #3
First draw a picture! Your picture should be of a circle (the sphere seen from the side) with a rectangle (the cylinder) inside it. If you set up a coordinate system with (0,0) at the center of the circle, you should be able to find either of h and r of the cylinder as a function of the other. (The Pythagorean theorem is helpful here.)
 

1. What is the formula for finding the maximum volume of a cylinder inscribed in a sphere of radius R?

The formula for finding the maximum volume of a cylinder inscribed in a sphere of radius R is V = (2/3)πR^3. This formula can be derived by using the Pythagorean theorem and the formula for the volume of a cylinder.

2. How do you find the minimum surface area of a cylinder inscribed in a sphere of radius R?

The minimum surface area of a cylinder inscribed in a sphere of radius R can be found by using the formula SA = 4πR^2. This formula can be derived by using the Pythagorean theorem and the formula for the surface area of a cylinder.

3. Is there a specific orientation of the cylinder that maximizes its volume when inscribed in a sphere of radius R?

Yes, the cylinder will have a maximum volume when its axis is parallel to the diameter of the sphere. This means that the height of the cylinder will be equal to the diameter of the sphere.

4. How does the volume of the inscribed cylinder change as the radius of the sphere increases?

The volume of the inscribed cylinder will increase as the radius of the sphere increases. This is because as the sphere gets larger, there is more space for the cylinder to fit inside, resulting in a larger volume.

5. Can this problem be applied to real-world scenarios?

Yes, this problem can be applied to various real-world scenarios, such as finding the maximum volume of a cylindrical water tank that can fit inside a spherical water tower of a given size. It can also be used in engineering and architecture to optimize the use of space in cylindrical structures.

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