1. The volume of a sphere is \(\cfrac{4}{3}\pi r^3\) cubic units, and the sphere is inscribed in a cube. The ratio of the volumes of the sphere and the cube is \(\pi : 6\).

2. The volume of a sphere is \(\frac{4}{3}\pi r^3\) cubic units; it is enclosed inside a cube. Find the ratio of the volumes of the cube and the sphere.

(a) \(3:\pi\) (b) \(4:\pi\) (c) \(6:\pi\) (d) \(8:\pi\)

3. What is the ratio of the volumes of a solid right circular cone and a solid sphere that can be carved with minimal wood wastage from two solid cubes, each having edge length \(a\)?

4. If a sphere occupies the maximum space inside a cube, what will be the ratio of the volumes of the sphere and the cube

(a) π:3 (b) π:2 (c) π:4 (d) π:6

5. If a hemisphere and a cone have equal base areas and equal heights, then what will be the ratio of their volumes?

(a) 1:3 (b) 2:3 (c) 3:2 (d) 3:1

6. A hemisphere and a cone have the same base and height. Determine the ratio of their volumes and the ratio of their curved surface areas.

7. A solid vertical cylindrical cone has both its diameter and height equal to 21 cm. Find the volume of the largest possible sphere that can be obtained from this cone. Also, determine the ratio of the volumes of the cone and the sphere.

8. What is the ratio of the volumes of a solid right circular cylinder, a solid right circular cone, and a solid sphere, all having the same diameter and the same height?

9. A solid hemisphere and a solid cone have the same base radius and height. What is the ratio of their volumes?

(a) 3:2 (b) 1:3 (c) 2:3 (d) None of the above

10. The volume of a sphere is directly proportional to the cube of its radius. If three solid spheres with radii of 3 cm, 4 cm, and 5 cm are melted to form a new solid sphere, and there is no loss in volume during melting, then find the radius of the new sphere.

11. A right circular cone and a hemisphere have equal base areas and volumes. Determine the ratio of their heights.

12. The volume of a sphere is directly proportional to the cube of its radius. Three solid spheres with diameters of \(1\frac{1}{2}\) m, 2 m, and \(2\frac{1}{2}\) m are melted and recast into a single solid sphere. Using the concept of proportionality, determine the diameter of the new sphere.

13. If a solid hemisphere and a solid cone have the same base radius and height, what is the ratio of their volumes?

(a) 3:2 (b) 1:3 (c) 2:3 (d) None of the above

14. A sphere is perfectly enclosed within a cube. What is the ratio of the volume of the sphere to the volume of the cube?

15. A solid right circular cone and a solid sphere have the same radius and the same volume. What is the ratio of the diameter of the sphere to the height of the cone?

(a) \(2 : 1 \) (b) \(1 : 2\) (c) \(1 : 1\) (d) \(2 : 3\)

16. Determine the volume ratio of a solid cone, a solid hemisphere, and a solid cylinder with equal base diameters and equal heights.

17. The radius of a hemisphere and the height of a cone are equal. If their volumes are numerically equal, then what is the ratio of the radius of the hemisphere to the radius of the cone?

18. A solid hemisphere and a solid cone have the same base radius and height. Determine the ratio of their volumes.

(a) 3:2 (b) 11:3 (c) 2:3 (d) 4:3

19. A solid hemisphere and a solid cone have the same base diameter and height. Determine the ratio of their volumes and the ratio of their curved surface areas.

20. A right circular drum has a radius of 21 cm and a height of 2i. A solid iron sphere has a diameter of 21 cm. Determine the ratio of the volumes of the drum and the solid iron sphere (neglecting the thickness of the drum). Now, if the drum is completely filled with water and the sphere is fully submerged and then removed, determine the new depth of the water in the drum.

21. The ratio of the volume of a cube to its inscribed sphere is—

(a) 3:π (b) 2:π (c) 6:π (d) π:6

22. What is the ratio of the volumes of a solid right circular cylinder, a solid right circular cone, and a solid sphere, all having the same diameter and height?

23. If the ratio of the total surface areas of two solid spheres is 1:4, what will be the ratio of their volumes? Calculate and write it down.

24. The volume of a sphere is in direct variation with the cube of its radius. If three solid spheres of radii 3 cm, 4 cm, and 5 cm are melted to form a single new solid sphere, and if there is no change in volume due to melting, then determine the diameter of the new sphere using the principle of variation.

25. We take a vertical cylindrical drum with radius 21 cm and height 21 cm, and a solid iron sphere with a diameter of 21 cm. We calculate the ratio of the volumes of the drum and the solid iron sphere (neglecting the thickness of the drum). Next, we completely fill the drum with water and then fully immerse the sphere into it. Now, determine the new depth of water in the drum after the sphere is removed.

26. A solid hemisphere and a solid cone have equal base diameters and equal heights. Calculate and write the ratio of their volumes and the ratio of their curved surface areas.

27. Determine the volume ratio of a solid cone, a solid hemisphere, and a solid cylinder that have equal base diameters and equal heights.

28. "The volume of a sphere is directly proportional to the cube of its radius. Three solid spheres with diameters of 1½, 2, and 2½ meters are melted to form a new solid sphere. Find the diameter of the new sphere. (Assume volume remains the same before and after melting.)

29. If a solid hemisphere and a solid cone have equal base areas and equal heights, then what is the ratio of their volumes?

(a) 1:3 (b) 1:2 (c) 2:3 (d) 3:4

30. If the numerical value of the volume and the surface area of a sphere are equal, what is the numerical value of the radius of the sphere?

(a) 3 (b) 5 (c) 7 (d) 4