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A cube of the 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 the maximum value (each corner touching the surface from ins...
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
Most Upvoted Answer
A cube of the maximum value (each corner touching the surface from ins...
Let's assume that the side length of the cube is $s$. Since the cube has a maximum value, each corner of the cube will touch the surface of the sphere from the inside. Therefore, the diagonal of the cube will be equal to the diameter of the sphere.
The diagonal of the cube can be found using the Pythagorean theorem. Let's consider one face of the cube. The diagonal of this face is the hypotenuse of a right triangle, with two sides of length $s$. Therefore, the diagonal of the cube is $\sqrt{2}s$.
Since the diagonal of the cube is equal to the diameter of the sphere, we have $\sqrt{2}s = 2r$, where $r$ is the radius of the sphere. Solving for $s$, we get $s = \frac{2r}{\sqrt{2}} = \sqrt{2}r$.
The volume of the cube is $s^3 = (\sqrt{2}r)^3 = 2\sqrt{2}r^3$.
The volume of the sphere is $\frac{4}{3}\pi r^3$.
The ratio of the volume of the cube to the volume of the sphere is $\frac{2\sqrt{2}r^3}{\frac{4}{3}\pi r^3} = \frac{3\sqrt{2}}{2\pi} = \frac{3\sqrt{2}\pi}{2\pi^2}$.
Therefore, the ratio of the volume of the cube to the volume of the sphere is $\boxed{\frac{3}{4}}$.
The diagonal of the cube can be found using the Pythagorean theorem. Let's consider one face of the cube. The diagonal of this face is the hypotenuse of a right triangle, with two sides of length $s$. Therefore, the diagonal of the cube is $\sqrt{2}s$.
Since the diagonal of the cube is equal to the diameter of the sphere, we have $\sqrt{2}s = 2r$, where $r$ is the radius of the sphere. Solving for $s$, we get $s = \frac{2r}{\sqrt{2}} = \sqrt{2}r$.
The volume of the cube is $s^3 = (\sqrt{2}r)^3 = 2\sqrt{2}r^3$.
The volume of the sphere is $\frac{4}{3}\pi r^3$.
The ratio of the volume of the cube to the volume of the sphere is $\frac{2\sqrt{2}r^3}{\frac{4}{3}\pi r^3} = \frac{3\sqrt{2}}{2\pi} = \frac{3\sqrt{2}\pi}{2\pi^2}$.
Therefore, the ratio of the volume of the cube to the volume of the sphere is $\boxed{\frac{3}{4}}$.
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