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A cube of maximum value (each corner touching the surface from inside) is cut from a sphere. What is the ratio of volume of the cube to that of the sphere?
- a)3 : 4π
- b)√3 : 2π
- c)2 : √3π
- d)4 : 3π
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
A cube of maximum value (each corner touching the surface from inside)...
When cube of maximum volume is cut from sphere, the diagonal of the cube = Diameter of the sphere
∴ √3 a = 2r
a/r = 2/√3
Volume of cube/Volume of sphere = a3/ (4/3 π r3)
⇒ 3/(4π) × (a/r)3
⇒ 3/(4π) × (2/√3)3
⇒ 2/(√3π)
∴ The required ratio = 2 ∶ √3π
∴ √3 a = 2r
a/r = 2/√3
Volume of cube/Volume of sphere = a3/ (4/3 π r3)
⇒ 3/(4π) × (a/r)3
⇒ 3/(4π) × (2/√3)3
⇒ 2/(√3π)
∴ The required ratio = 2 ∶ √3π
Most Upvoted Answer
A cube of maximum value (each corner touching the surface from inside)...
Volume of Cube and Sphere:
- The volume of a cube is given by V_cube = s^3, where s is the side length of the cube.
- The volume of a sphere is given by V_sphere = (4/3)πr^3, where r is the radius of the sphere.
Relationship between Cube and Sphere:
- The cube is inscribed in the sphere, touching the sphere at each corner from inside.
- The diameter of the sphere is equal to the diagonal of the cube, which is √3 times the side length of the cube.
Finding the Ratio of Volumes:
- Let the side length of the cube be s, and the radius of the sphere be r.
- Since the cube is inscribed in the sphere, the diagonal of the cube is equal to the diameter of the sphere: s√3 = 2r.
- Solving for s in terms of r, we get s = 2r / √3.
- Substituting this into the volume of the cube formula, V_cube = (2r / √3)^3 = (8/3√3)r^3.
- Now, we can find the ratio of the volume of the cube to the volume of the sphere:
V_cube / V_sphere = (8/3√3)r^3 / (4/3)πr^3
V_cube / V_sphere = 2 / √3π
Therefore, the ratio of the volume of the cube to the volume of the sphere is 2 : √3π, which corresponds to option 'C'.
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