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A cube of maximum possible volume is kept inside a sphere. A right circular cylinder of maximum possible volume is kept inside the cube. What is the ratio of the total outer surface area of the cylinder to that of the sphere?
- a)1 : 2
- b)1 : 3
- c)1 : 4
- d)1 : 5
Correct answer is option 'A'. Can you explain this answer?
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A cube of maximum possible volume is kept inside a sphere. A right cir...
Understanding the Problem
To find the ratio of the total outer surface area of a cylinder to that of a sphere, we need to analyze the geometric relationships between the cube, sphere, and cylinder.
Volume of the Sphere
- Let the radius of the sphere be R.
- The maximum volume of a cube that can fit inside the sphere has its diagonal equal to the diameter of the sphere.
- The side length (a) of the cube can be derived using the relationship: a√3 = 2R, giving us a = (2R)/√3.
Volume of the Cylinder
- The maximum volume of a right circular cylinder that can fit inside the cube has its height equal to the side length of the cube (a) and its diameter equal to the side length of the cube.
- The radius (r) of the cylinder, therefore, is a/2 = (R/√3).
Surface Area Calculations
- The total surface area (A) of the cylinder is given by: A_cylinder = 2πr(h + r) = 2π(R/√3)((2R)/√3 + (R/√3)) = 2π(R/√3)(3R/√3) = 2π(R^2).
- The total surface area (A) of the sphere is given by: A_sphere = 4πR^2.
Finding the Ratio
- The ratio of the surface area of the cylinder to the surface area of the sphere is: A_cylinder : A_sphere = 2π(R^2) : 4π(R^2) = 1 : 2.
Conclusion
Thus, the ratio of the total outer surface area of the cylinder to that of the sphere is 1 : 2, confirming that the correct answer is option 'A'.
To find the ratio of the total outer surface area of a cylinder to that of a sphere, we need to analyze the geometric relationships between the cube, sphere, and cylinder.
Volume of the Sphere
- Let the radius of the sphere be R.
- The maximum volume of a cube that can fit inside the sphere has its diagonal equal to the diameter of the sphere.
- The side length (a) of the cube can be derived using the relationship: a√3 = 2R, giving us a = (2R)/√3.
Volume of the Cylinder
- The maximum volume of a right circular cylinder that can fit inside the cube has its height equal to the side length of the cube (a) and its diameter equal to the side length of the cube.
- The radius (r) of the cylinder, therefore, is a/2 = (R/√3).
Surface Area Calculations
- The total surface area (A) of the cylinder is given by: A_cylinder = 2πr(h + r) = 2π(R/√3)((2R)/√3 + (R/√3)) = 2π(R/√3)(3R/√3) = 2π(R^2).
- The total surface area (A) of the sphere is given by: A_sphere = 4πR^2.
Finding the Ratio
- The ratio of the surface area of the cylinder to the surface area of the sphere is: A_cylinder : A_sphere = 2π(R^2) : 4π(R^2) = 1 : 2.
Conclusion
Thus, the ratio of the total outer surface area of the cylinder to that of the sphere is 1 : 2, confirming that the correct answer is option 'A'.
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A cube of maximum possible volume is kept inside a sphere. A right circular cylinder of maximum possible volume is kept inside the cube. What is the ratio of the total outer surface area of the cylinder to that of the sphere?a)1 : 2b)1 : 3c)1 : 4d)1 : 5Correct answer is option 'A'. Can you explain this answer?
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