1. 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

2. 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.

3. 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) \(8:\pi\)

4. 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 \(a\)?

5. If the ratio of the volumes of two spheres is 27:8, what will be the ratio of their curved surface areas?

6. 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 \(\pi : 6\).

7. If the volumes of two solid spheres are in the ratio 27:8, what will be the ratio of their curved surface areas?

(a) 1:2 (b) 9:4 (c) 1:8 (d) 1:16

8. The volume ratio of a sphere and the circumscribed cube of the sphere is

(a) π:2 (b) π:3 (c) π:4 (d) π:6

9. If the volumes of two solid right circular cylinders are equal and their heights are in the ratio 1:2, what will be the ratio of the lengths of their radii?

(a) 1:√2 (b) √2:1 (c) 1:2 (d) 2:1

10. The radii of two solid iron spheres are \(r_1\) and \(r_2\) respectively. The spheres are melted and recast into a single solid sphere. What will be the radius of the resulting sphere?

(a) \((r_1^3+r_2^3)\) (b) \((r_1^3+r_2^3)^3\) (c) \((r_1+r_2)^3\) (d) \((r_1^3+r_2^3)^{\cfrac{1}{3}}\)

11. If the radii of two cones are in the ratio 2:3 and their heights are in the ratio 5:3, then what will be the ratio of their volumes?

(a) 4:9 (b) 9:4 (c) 27:20 (d) 20:27

12. 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.

13. 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?

14. If the ratio of the volumes of two cubes is 8:27, then what will be the ratio of their total surface areas?

15. What is the ratio of the surface area of a cube to the curved surface area of the largest sphere that can be inscribed within it?

16. 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

17. If the volumes of two cones are in the ratio \(1:4\) and their radii are in the ratio \(4:5\), then what will be the ratio of their heights?

(a) 1:5 (b) 5:4 (c) 5:16 (d) None of the above

18. 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

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

20. The ratio of the total surface area of a sphere and a hemisphere with a unit radius will be 4:3.

21. 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?

22. The ratio of the base radii of a right circular cylinder and a right circular cone is 3:4, and the ratio of their heights is 2:3. What will be the ratio of the volumes of the cylinder and the cone?

23. If a solid hemisphere is melted to form a sphere, what will be the ratio of their radii?

24. If the ratio of the radii of two right circular cylinders is 2:3 and the ratio of their heights is 5:6, what will be the ratio of their volumes?

25. If three solid spheres with radii of 3 cm, 4 cm, and 5 cm are melted and reformed into a larger sphere, what will be the radius of the new sphere?

26. 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?

27. A hollow sphere is made of lead sheet 1 cm thick and has an outer radius of 6 cm. If the sphere is melted and recast into a right circular cylinder with a radius of 2 cm, what will be the length of the cylinder? Let me know if you'd like the full solution too. I'm ready when you are.

28. "The volume of a sphere varies directly with the cube of its radius, and the surface area of a sphere varies directly with the square of its radius. Prove that the square of the volume of a sphere will be directly proportional to the cube of its surface area."

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. The ratio between the length of a cube’s space diagonal and the diagonal of one of its faces will be ----.