Question

HELP PLS? A sphere with radius 8 cm is inscribed in a cube. Find the ratio of the volume of the cube to the volume of the sphere​

Answer 1

`\sf Ratio= (6)/(\pi)`

Explanation:

To find the ratio of the volume of the cube to the volume of the inscribed sphere, we can simply use the volume formulas and the fact that the diameter of the sphere is equal to the side length of the cube.

Given that the diameter (d) is twice the radius (r), it follows that the side length (s) of a cube is equal to 2r:

`s = 2r`

The formula for the volume of a cube with side length s is V = s³.

Therefore, if s = 2r, then the formula for the volume of the cube in this scenario is:

`\sf V_(cube) = (2r)^3`

`\sf V_(cube) = 2^3 \cdot r^3`

`\sf V_(cube) = 8r^3`

The formula for the volume of a sphere is given by:

`\sf V_(sphere) = (4)/(3)\pi r^3`

So, the ratio of the volume of the cube to the volume of the inscribed sphere is:

`\sf Ratio=(V_(cube))/(V_(sphere))`

`\sf Ratio= (8r^3)/((4)/(3)\pi r^3)`

`\sf Ratio= (8)/((4)/(3)\pi )`

`\sf Ratio= (8\cdot 3)/(4\pi)`

`\sf Ratio= (24)/(4\pi)`

`\sf Ratio= (6)/(\pi)`

Therefore, the ratio of the volume of a cube to the volume of an inscribed sphere is 6/π.