1. If the radius of a right circular cone is halved and its height is doubled, then the volume of the cone will be — of its original volume.

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

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

4. 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.

5. 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, then write the ratio of the height of the cone to the radius of its base.

6. If the ratio of the volume of a sphere with radius ???? to the volume of a cone with the same radius is 4:9, then the height of the cone will be..

(a) \(3r\) (b) \(4r\) (c) \(6r\) (d) \(9r\)

7. 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.

8. The diameter of one sphere is twice the diameter of another sphere. If the numerical value of the surface area of the larger sphere is equal to the numerical value of the volume of the smaller sphere, then what is the radius of the smaller sphere?

9. A vertical circular cylinder has a height that is twice its radius. If the height were six times the radius, then the volume of the cylinder would be 539 cubic decimeters more. What is the height of the cylinder?

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

11. 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.

12. If the radius of a right circular cylinder is 2 units, then for any height, the numerical values of the cylinder’s volume and curved surface area will be equal.

13. If the volume of a cone with a height of 12 cm is 154 cubic cm, then what will be the radius of its base?

(a) 3.5 cm (b) 4.5 cm (c) 5 cm (d) 5.5 cm

14. If both the radius and the height of a cone are doubled, then the volume of the new cone will be how many times the volume of the original cone?

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

16. The surface area of a sphere is equal in square units to twice its volume in cubic units. Find the radius of the sphere.

17. The height of a right circular cylinder is twice its radius. If the height were six times its radius, then the volume of the cylinder would be 539 cubic decimeters more. Determine the volume of the cylinder.

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

19. If the radius of a right circular cylinder is halved and the height is doubled, the volume of the new cylinder will be what fraction of the original cylinder’s volume?

(a) equal (b) double (c) half (d) Four times.

20. If the radius of a right circular cylinder is 2 units, then for any height, the numerical values of the cylinder’s volume and its curved surface area will be equal.

21. 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.

22. "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.)

23. From a solid hemisphere with radius \( r \) units, the volume of the largest solid cone that can be cut out will be _____ .

24. If the radius and height of a cone are each doubled, what will be the volume of the new cone in relation to the volume of the original cone?

(a) 3 times (b) 4 times (c) 6 times (d) 8 times

25. The volume of a right circular cone is V. If both the height and the radius are doubled, the new volume will be:

(a) 3V (b) 4V (c) 6V (d) 8V

26. 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

27. 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

28. The diameter of one sphere is twice the diameter of another sphere. The curved surface area of the first sphere is equal to the volume of the second sphere. What is the radius of the first sphere?

(a) 21 (b) 22 (c) 23 (d) 24

29. The volume of a sphere with radius \(\cfrac{r}{2}\) units will be \(\cfrac{4}{3}\pi\left(\cfrac{r}{2}\right)^3 = \cfrac{\pi r^3}{6}\).

(a) \(\cfrac{1}{6}\pi r^3\) cubic units (b) \(\cfrac{4}{3}\pi r^3\) cubic units (c) \(\cfrac{2}{3}\pi r^3\)cubic units (d) \(\cfrac{1}{3}\pi r^3\)cubic units

30. 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.