1. A hollow cylinder with a height of 13 cm has an outer radius of 6.75 cm and an inner radius of 5.25 cm. It is melted to form a solid cylinder with a height of 6.5 cm. Find the radius of the solid cylinder.

2. A hollow iron cylinder of height 20 cm has an outer radius of 5 cm and an inner radius of 4 cm. This cylinder is melted and recast into a solid cone whose height is one-third of the original cylinder's height. Find the diameter of the base of the cone.

3. A hollow right circular cylinder has an outer diameter of 16 cm and an inner diameter of 12 cm. The height of the cylinder is 36 cm. If it is melted down, how many solid cylinders of diameter 2 cm and height 5 cm can be made from it?

4. 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.

5. A vertical hollow cylindrical pipe has an outer radius of 16 cm and an inner radius of 12 cm, with a height of 36 cm. If the pipe is melted and solid cylindrical rods are made from it, each with a diameter of 2 cm and a length of 6 cm, determine how many such solid rods can be produced.

6. 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.

7. 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."

8. 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.

9. 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.

10. 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

11. To make a hollow cylindrical pipe of uniform thickness, a solid iron sphere of radius 6 cm is melted. The pipe has a length of 18 cm and an outer radius of 10 cm. Find the thickness of the pipe.

12. A solid cylindrical iron rod has a base radius of 32 cm and a height of 35 cm. If the rod is melted and recast into solid cones, each with a radius of 8 cm and a height of 28 cm, how many such cones can be made?

13. The height of a solid right circular cone is 20 cm and its slant height is 25 cm. If a solid right circular cylinder of height 15 cm has the same volume as the cone, then calculate and write the diameter of the base of the cylinder.

14. The radius of a solid right circular iron rod is 32 cm and its length is 35 cm. If the rod is melted and recast into solid cones each with a radius of 4 cm and height of 28 cm, calculate how many such cones can be made.

15. A solid right circular cone is melted and recast into a solid right circular cylinder with the same radius and a height of 5 cm. Find the height of the cone.

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

16. A hemispherical bowl made of silver foil has an outer diameter of 4 cm and an inner diameter of 4 cm. The bowl is melted down to form a solid cone with a diameter of 4 cm. Find the height of the cone."

17. A right circular cylinder made of iron sheet is open at both ends. Its height is 42 cm, thickness is 1 cm, and its outer radius is 10 cm. Find the amount of iron used to make the cylinder.

18. 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.

19. 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?

20. A solid right circular cone is melted and recast into a solid right circular cylinder. The radius of both the cone and the cylinder is the same. If the height of the cone is 15 cm, calculate and write the height of the cylinder.

21. Two solid spheres with radii 1 cm and 6 cm are melted to form a hollow sphere with an outer radius of 9 cm. Find the inner radius of the hollow sphere.

22. A vertical hollow cylindrical pipe made of iron has both ends open. Its height is 35 cm, the outer radius is 12 cm, and the inner radius is 10 cm. What is the volume of iron in the pipe in cubic centimeters?

23. Two solid spheres with diameters of 20 cm and 6 cm are melted down, and the material is used to make a hollow sphere with a thickness of 1 cm. What is the inner radius of the hollow sphere.

24. A hollow vertical cylindrical iron pipe has an outer radius of 5 cm and an inner radius of 4 cm. If the total surface area of the pipe is 1188 square cm, what is the length of the pipe?

25. 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.

26. A solid cylindrical iron rod has a cross-sectional diameter of 16 cm and a length of 1 meter. If this rod is melted and used to form right circular cones, each with a height of 4 cm and a base radius of 5 cm, how many such cones can be made?

27. A sphere and a right circular cylinder both have a radius of 3 cm. If their volumes are equal, what is the height of the cylinder?

28. 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

29. A solid lead sphere has a diameter of 14 cm. If this sphere is melted and reshaped into solid spheres with a radius of 3.5 cm, how many such spheres can be formed?

30. Two solid spheres with radii of 1 cm and 6 cm are melted to form a completely hollow sphere with a radius of 5 cm. Determine the volume and curved surface area of the new sphere.