Answer the questions independently of each other.The largest possible sphere is kept inside a cube. Then eight identical spheres of maximum possible volume are inserted inside the cube near the corners touching the large sphere and three sides of the cube. What is the ratio of the volumes of the larger sphere and one of the smaller spheres?

To solve this problem, we need to understand the geometry of the situation.

1. The diameter of the larger sphere is equal to the side of the cube. Let's denote this length as 'a'. Therefore, the radius of the larger sphere is 'a/2'.

2. The smaller spheres are placed in such a way that they touch the larger sphere and three sides of the cube. This means that the diameter of a smaller sphere is equal to the space left in the side of the cube after accommodating the radius of the larger sphere. So, the diameter of the smaller sphere is 'a - a/2 = a/2'. Therefore, the radius of the smaller sphere is 'a/4'.

3. The volume of a sphere is given by the formula (4/3)πr³.

4. Therefore, the volume of the larger sphere is (4/3)π(a/2)³ = (4/3)π(a³/8) and the volume of the smaller sphere is (4/3)π(a/4)³ = (4/3)π(a³/64).

5. The ratio of the volumes of the larger sphere to a smaller sphere is therefore (4/3)π(a³/8) : (4/3)π(a³/64) = 8:1.

So, the ratio of the volumes of the larger sphere to one of the smaller spheres is 8:1.

To solve this problem, we need to understand the geometry of the situation.

1. The diameter of the larger sphere is equal to the side of the cube. Let's denote this length as 'a'. Therefore, the radius of the larger sphere is 'a/2'.

2. The smaller spheres are placed in such a way that they touch the larger sphere and three sides of the cube. This means that the diameter of a smaller sphere is equal to the space left in the side of the cube after accommodating the radius of the larger sphere. So, the diameter of the smaller sphere is 'a - a/2 = a/2'. Therefore, the radius of the smaller sphere is 'a/4'.

3. The

To solve this problem, we need to understand the geometry of the situation.

1. The diameter of the larger sphere is equal to the side of the cube. Let's denote this length as 'a'. So, the radius of the larger sphere is 'a/2'.

2. The smaller spheres are placed in such a way that they touch the larger sphere and three sides of the cube. This means that the diameter of a smaller sphere is equal to the side of the cube minus the diameter of the larger sphere, which is 'a - a = 0'. However, this is not possible.

3. The correct interpretation is that the smaller spheres touch the larger sphere and the cube at the corners. This means that the diameter of a smaller sphere is equal to the side of the cube divided by 2, which is 'a/2'. So, the radius of a smaller sphere is 'a/4'.

4. The volume of a sphere is given by the formula V = 4/3 * π * r^3.

5. Therefore, the volume of the larger sphere is V1 = 4/3 * π * (a/2)^3 = πa^3/6.

6. The volume of a smaller sphere is V2 = 4/3 * π * (a/4)^3 = πa^3/96.

7. The ratio of the volumes of the larger sphere to a smaller sphere is V1/V2 = (πa^3/6) / (πa^3/96) = 96/6 = 16.

So, the ratio of the volumes of the larger sphere to one of the smaller spheres is 16:1.