1. If a solid sphere is melted to form a solid right circular cylinder, then the _____ of the sphere and the cylinder are equal.

2. If a right circular cylinder with radius \(2x\) cm and height \(x\) cm is melted to form a sphere, then the radius of the resulting sphere will be –

(a) \(\sqrt[3]3{x}\) cm (b) \(\sqrt[3]{3x}\) (c) \(x\) (d) \(2x\)

3. 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.

4. A solid right circular cylinder and two hemispheres have equal base radii. If the two hemispheres are attached to the flat circular ends of the cylinder, then the total surface area of the resulting solid = curved surface area of one hemisphere + curved surface area of the cylinder + curved surface area of the other hemisphere.

5. A solid sphere is melted and recast into a solid right circular cylinder. The volumes of the sphere and the cylinder are_______.

6. 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

7. 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.

8. A solid sphere is melted to form a solid vertical cylindrical rod. The volume of the sphere and the cylinder is _____.

9. A rectangle has a length of \( a \) units and a breadth of \( b \) units. When the rectangle is wrapped to form a right circular cylinder, the circumference of the base of the cylinder is equal to the length of the rectangle. The lateral surface area of the cylinder will be _____.

10. A hollow sphere is made of lead sheet 1 cm thick and has an outer radius of 6 cm. If the sphere is melted and recast into a right circular cylinder with a radius of 2 cm, what will be the length of the cylinder? Let me know if you'd like the full solution too. I'm ready when you are.

11. The English translation is: "A solid gold sphere with a radius of 4.2 cm is melted and recast into a solid right circular cylinder with a radius of 2.8 cm. Determine the height (length) of the cylinder."

12. 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.

13. 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

14. 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.

15. 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?

16. A solid sphere has a surface area of 616 square cm. It is melted to form 14 identical right circular cones, each with a height of 2 cm. Find the diameter of the base of each cone.

17. If the volumes of two solid right circular cylinders are equal and their height ratio is 1:4, then their radius ratio will be—.

(a) \(2:1\) (b) \(\sqrt2:1\) (c) \(1:2\) (d) \(1:\sqrt2\)

18. If a solid hemisphere is melted to form a sphere, what will be the ratio of their radii?

19. A solid silver sphere with a diameter of 6 cm is melted to form a right circular cone with a height of 3 cm. Determine the diameter of the cone.

20. If the volumes of two solid right circular cylinders are equal and their height ratio is 1:2, then their radius ratio will be –?

21. 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.

22. A solid rod has a length of \( h \) meters and a diameter of \( r \) meters. It is melted to form 6 spheres, each with a radius of \( r \) meters. Determine the relationship between \( h \) and \( r \).

23. If three solid spheres with radii of 3 cm, 4 cm, and 5 cm are melted and reformed into a larger sphere, what will be the radius of the new sphere?

24. 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 height?

25. A hollow sphere made of lead sheets with a thickness of 1 cm has an outer radius of 6 cm. If the sphere is melted to form a solid cylindrical rod with a radius of 2 cm, what will be the length of the rod?

26. The curved surface area of a solid sphere is equal to the curved surface area of a solid right circular cylinder. The height and diameter of the cylinder are both 12 cm. Write the radius of the sphere.

27. If a solid sphere and a solid right circular cylinder have equal radii and the same volume, then calculate and write the ratio of the radius to the height of the cylinder.

28. A solid silver sphere with a diameter of 6 decimeters is melted and recast into a solid right circular rod that is 1 decimeter long. Calculate and write the diameter of the rod.

29. The radius of the cross-section of a solid right circular rod is 3.2 decimeters. The rod was melted and recast into 21 solid spheres, each with a radius of 4 centimeters. If so, calculate and write the length of the rod.

30. A hollow sphere made of lead sheets with a thickness of 1 cm has an outer radius of 6 cm. If this sphere is melted and recast into a solid right circular rod with a radius of 2 cm, calculate the length of the rod.