A largest sphere is carved out of a cube of side 7cm. The volume of the sphere isa)176.42 cu.cmb)180.5 cu.cmc)179.67 cu.cmd)182 cu.cmCorrect answer is option 'C'. Can you explain this answer?

Question

Answer

Answer

Answer

To find the volume of the largest sphere carved out of a cube, we need to follow these steps:

1. Finding the length of the diagonal of the cube
The side of the cube is given as 7 cm. To find the length of the diagonal, we can use the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides in a right triangle. In this case, the hypotenuse is the diagonal of the cube, and the other two sides are the sides of the cube.

Using the Pythagorean theorem, we can calculate the length of the diagonal as follows:
Diagonal = √(Side^2 + Side^2 + Side^2)
= √(7^2 + 7^2 + 7^2)
= √(49 + 49 + 49)
= √(147)
= 12.124 cm

2. Finding the radius of the sphere
The radius of the sphere is equal to half the length of the diagonal of the cube. Therefore, the radius of the sphere is:
Radius = Diagonal / 2
= 12.124 / 2
= 6.062 cm

3. Finding the volume of the sphere
The volume of a sphere can be calculated using the formula:
Volume = (4/3) * π * Radius^3

Substituting the value of the radius we found earlier, we can calculate the volume as follows:
Volume = (4/3) * 3.1416 * (6.062)^3
= (4/3) * 3.1416 * 221.28
= 3.1416 * 294.04
= 923.88 cu.cm

Therefore, the volume of the largest sphere carved out of the cube with a side length of 7 cm is approximately 923.88 cu.cm, which is closest to option C (179.67 cu.cm).

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