1. 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

2. A cone and a cylinder with the same base radius and the same height will have the cone's volume equal to half the volume of the cylinder.

3. 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

4. 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) \(2 : 3\)

5. 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?

6. 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.

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

8. 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.

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 cone is jointly proportional to the square of the radius of the base and its height. If the ratio of the radii of two cones is 2:3 and their heights are in the ratio 5:4, then find the ratio of their volumes.

11. 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

12. 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

13. 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.

14. A right circular cylinder with the same radius and height has a volume that is three times the volume of a right circular cone.

15. If a cylinder and a sphere have the same volume with equal radius length, determine the ratio of the cylinder's diameter to its height.

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

17. 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

18. 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

19. The radius of the base of a right circular cone is equal to the radius of a sphere. The volume of the cone (in cubic centimeters) is equal to the curved surface area of the sphere (in square centimeters). What is the height of the cone?

(a) 4 cm (b) 3 cm (c) 8 cm (d) 12 cm

20. What is the ratio of the volumes of a right circular cylinder and a cone having the same radius and height?

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

21. Two solid spheres with radii of 8 cm and 10 cm are melted to form a right circular cone with a height of 42 cm. What is the radius of the cone?

(a) 8 cm (b) 10 cm (c) 12 cm (d) 16 cm

22. A sphere and a right circular cone have the same radius \(r\) and equal volume. What is the height of the cone?

(a) \(\cfrac{r}{3}\) (b) \(\cfrac{r}{4}\) (c) \(\cfrac{3r}{4}\) (d) \(\cfrac{4r}{3}\)

23. A right circular cone and a right circular cylinder with the same base and the same height will have a volume ratio of 1 : 3.

24. 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.

25. 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.

26. A solid right circular cone is melted to form a solid right circular cylinder of the same radius, with a height of \(5\) cm. Determine the height of the cone.

(a) 10 cm (b) 15 cm (c) 18cm (d) 24cm

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

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

29. A solid right circular cone is melted to form a solid cylinder of the same radius, with a height of 5 cm. Find the height of the cone.

30. If the height of a right circular cone remains the same and its radius is doubled, the volume of the cone will increase by 300%.