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 numerical values of the volume and curved surface area of a sphere are equal, then its radius is _____ units.

3. 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?

4. If the numerical values of the volume and the curved surface area of a right circular cylinder are equal, then the diameter of the cylinder is ___________ units.

5. 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

6. If the volume and curved surface area of a sphere are numerically equal, what is its radius?

(a) 5 units (b) 3 units (c) 2 units (d) 7 units

7. If the curved surface area of a solid sphere is \(s\) and its volume is \(v\), then what is the value of \(\frac{s^3}{v^2}\)?

8. If the numerical value of the curved surface area and the volume of a sphere are equal, then the radius will be 3 units.

9. The volume and total surface area of a solid hemisphere are numerically equal. Then the diameter of the hemisphere will be –

(a) 4:5 units (b) 6 units (c) 9 units (d) 3 units

10. If the curved surface area of a sphere is \(S\) square units and its volume is \(V\) cubic units, then determine the relationship between \(S\) and \(V\).

11. 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.

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

13. If the curved surface area of a solid hemisphere is \( S \) and its volume is \( V \), find the value of \( \cfrac{S^3}{V^2} \).

14. If the radius of a solid hemisphere is 2r units, then the total surface area is _______ \( πr^2 \) square units.

15. If the curved surface area of a solid sphere is \(S\) and the volume is \(V\), then write the value of \( \cfrac{S^3}{V^2} \) (without substituting the value of \(π\)).

16. 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.

17. 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.

18. If the surface area of a solid sphere is \(S\), and its volume is \(V\), then find the value of \(\cfrac{S^3}{V^2}\), without substituting the value of \(\pi\).

19. If a right circular cone and a sphere have the same radius, and the numerical value of the cone's volume is equal to the numerical value of the sphere's curved surface area, then what is the height of the cone?

(a) 10 units (b) 11 units (c) 12 units (d) 13 units

20. The length of a right rectangular prism is three times its width and five times its height. If its volume is 14,400 cubic centimeters, then what is its total surface area?

(a) 4300 square cm (b) 4320 square cm (c) 4500 square cm (d) 4520 square cm

21. If the numerical values of the curved surface area and the total surface area of a sphere are equal, what is its radius?

(a) 3 units (b) 4 units (c) 5 units (d) 6 units

22. If the surface area of a sphere is \(A\) and its volume is \(V\), then find the value of \(\cfrac{A^3}{V^2}\).

23. If the numerical value of the volume and the curved surface area of a right circular cone are equal, find the length of the diameter of the cone — in units.

24. 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?

25. 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.

26. 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.

27. The total surface area and volume of a solid hemisphere are numerically equal. Find the radius of the base of the hemisphere.

28. If a solid hemisphere and a solid cone have equal base diameters and equal heights, find the ratio of their curved surface areas.

29. If the total surface area and the volume of a cube are numerically equal, then the length of its edge will be -- units.

30. If two solid spheres with diameters of 1 cm and 6 cm are melted and recast into a hollow sphere with a thickness of 1 cm, then what is the outer surface area of the new hollow sphere?