1. 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.

2. 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

3. 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) \(2 : 3\)

4. A solid object has its lower part in the shape of a hemisphere and its upper part in the shape of a right circular cone. If the surface areas of both parts are equal, determine the ratio of the radius to the height of the cone.

5. A solid sphere with a radius of \(r\) units is melted and recast into a solid right circular cone with a height of \(r\) units. Find the radius of the base of the cone.

(a) 2r units (b) 3r units (c) r unit (d) 4r units

6. The base radius of a solid right circular cone is \(r\) units, its vertical height is \(h\) units, and its slant height is \(l\) units. The cone’s base is attached to the base of a solid right circular cylinder. If the base radius and height of the cylinder are equal to those of the cone, then the total surface area of the combined solid is \((πrl + 2πrh + 2πr^2)\) square units.

7. 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.

8. The height of a right circular solid cone is twice its radius. If the height were increased sixfold, the volume of the cone would be 539 cubic centimeters more. Find the height of the cone.

9. The bottom part of a solid is in the shape of a hemisphere, and the top part is in the shape of a right circular cone. If the surface areas of both parts are equal, then calculate and write the ratio of the radius to the height of the cone.

10. 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

11. The lateral surface area of a right circular cone is \( \sqrt{5} \) times the area of its base. What is the ratio of the cone’s height to the radius of its base?

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

13. 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.

14. If a solid sphere and a solid right circular cylinder have the same radius and equal volume, then write the ratio of the cylinder's radius to its height.

15. The slant height of a right circular cone is 7 cm, and its total surface area is 147.84 square cm. Find the radius of its base.

16. The curved surface area of a right circular cone is √5 times the area of its base. What is the ratio of the cone’s height to its radius?

17. 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

18. If a solid cone and a solid right circular cylinder have the same base radius and same height, and the ratio of their curved surface areas is 5:8, then determine the ratio of their base radius to height.

19. "The lateral surface area of a right circular cone is √5 times the area of its base. Write the ratio of the height to the radius of the cone."

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

21. The height of a right circular cone is twice the length of its radius. If the height were 7 times the base diameter, the volume of the cone would be 539 cubic cm more. Find the height of the cone.

22. A solid sphere has a total surface area of 1386 cm². It is melted and reshaped into several right circular cones, each with a radius of 3.5 cm and a height of 6 cm. Find the radius of the solid sphere and determine how many such cones are formed.

23. 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

24. The numerical value of the volume and the total surface area of a solid hemisphere are equal. Find the radius of the hemisphere.

25. A hollow metallic sphere has an inner radius of 3 cm and an outer radius of 5 cm. It is melted to form a solid right circular cylinder of height \(\frac{8}{3}\) cm. Find the diameter of the base of the cylinder.

26. 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

27. If the numerical value of the volume of a right circular cone is equal to the numerical value of its curved surface area, what is the radius of the cone?

28. A right circular cylinder has a height-to-base-radius ratio of 3:1. If the volume of the cylinder is \(1029\pi\) cubic cm, find the total surface area of the cylinder.

29. 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.

30. The height of a right circular cone is twice its radius. If the height were 6 times the radius instead, the volume of the cone would be 539 cubic centimeters more. Find the height of the cone.