1. If the curved surface area of a solid sphere is numerically equal to three times its volume, then the numerical value of the diameter of the sphere is —.

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

2. If the radius of a solid sphere is doubled, then its volume will become three times.

3. If the curved surface area of a solid sphere and three times its volume are numerically equal, then the diameter of the sphere is _____ units.

4. If the height of a right circular cone remains unchanged and its radius is increased by 1 unit, the volume of the cone becomes four times its original volume. What is the length of the diameter of the cone?

5. 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.

6. If the radius of a solid sphere is doubled, its volume will also double.

7. 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.

8. The radius of a right circular cylinder is \(r\) units, and its height \(R\) units is \(\frac{2}{3}\) times the diameter of a sphere of radius \(R\). If their volumes are equal, show that \(r = R\).

9. If the diameter of a solid sphere is tripled, its surface area will be _____ times larger.

10. If the numerical values of the volume and surface area of a solid sphere are equal, then what is the numerical value of its diameter?

11. If the curved surface area of a solid sphere is numerically equal to three times its volume, find the radius of the sphere.

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

12. If the radius of a solid sphere is doubled, its volume will become twice as much.

13. 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.

14. "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.)

15. The volume of the largest solid cone that can be cut out from a solid hemisphere with a radius of r units.

(a) \(4\pi r^3\) cubic units. (b) \(43\pi r^3\) cubic units. (c) \(\cfrac{πr^3}{4} \)  cubic units (d) \(\cfrac{πr^3}{3}\) cubic units

16. If the total surface area of a solid sphere is 462 square centimeters, what is its diameter?

(a) 7 cm (b) 14 cm (c) 21 cm (d) 3.5 cm

17. 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?

18. The radius of the base of a solid right circular cone is equal to the radius of a solid sphere. If the volume of the sphere is twice the volume of the cone, find the ratio of the height of the cone to the radius of its base.

19. A solid sphere and a solid cone have the same diameter, and their total surface areas are equal. What is the ratio of the cone's height to its diameter?

20. If the radius of a sphere is doubled, then its volume will be twice the volume of the original sphere.

21. If the volume of a solid hemisphere is \(144π\) cubic centimeters, what is the diameter of the sphere?

22. 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.

23. The volume of a sphere is directly proportional to the cube of its radius. A lead sphere has a radius of 14 cm. Using the concept of proportion, prove that this sphere can be melted to form four spheres, each with a radius of 7 cm. (Assume that the volume remains constant during melting.)

24. 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) \(2 : 3\)

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

26. 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.

27. 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.

28. If the volume of a solid hemisphere is \(144\pi\) cubic cm, what is its total surface area?

29. Determine the volume of the largest solid cone that can be cut from a solid hemisphere with a radius of \(r\) units.

30. If both the radius and height of a right circular cylinder are doubled, its volume will be four times the original volume.