1. 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.

2. 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?

3. Translate the following: A solid iron sphere of radius 8 cm is melted and recast into solid spherical balls, each of radius 1 cm. Determine how many such balls can be made.

4. A hemispherical vessel with an inner radius of 9 cm is completely filled with water. This water is poured into cylindrical bottles, each having a diameter of 3 cm and a height of 4 cm. How many such bottles are required to hold all the water?

5. 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?

6. 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.

7. "A hollow right circular cylinder has an external diameter of 16 cm and an internal diameter of 12 cm. Its height is 36 cm. If it is melted and recast into solid cylinders, each with a diameter of 2 cm and a height of 6 cm, calculate and write how many such solid cylinders can be made."

8. A hemispherical container with an inner radius of 9 cm is completely filled with water. This water is to be transferred into conical bottles, each having a diameter of 3 cm and a height of 4 cm. Calculate how many such bottles are needed to empty the container.

9. 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.

10. A solid sphere with a radius of 4.2 cm is melted and recast into a solid cylindrical rod with a diameter of 2.8 cm. Find the length of the rod.

11. A solid lead sphere with a diameter of 12 cm is melted down to form three smaller solid spheres. If the diameters of the smaller spheres are in the ratio 3 : 4 : 5, then what is the radius of each of the smaller spheres?

12. A solid iron sphere of radius 8 cm is melted down to form solid balls, each of radius 1 cm. Calculate and write how many such balls can be made.

13. A solid copper cuboid of length 6.6 decimeters, width 4.2 decimeters, and thickness 1.4 decimeters is melted and recast into solid spheres, each with a diameter of 2.1 decimeters. Calculate how many such spheres can be made and how much metal will be in each sphere (in cubic decimeters).

14. A solid lead sphere with a diameter of 12 cm is melted to form three smaller solid lead spheres. If the diameters of the smaller spheres are in the ratio 3 : 4 : 5, find the radius of each smaller sphere.

15. A solid lead sphere with a diameter of 12 cm is melted to form three smaller spheres. If the diameters of the smaller spheres are in the ratio 3:4:5, what is the radius of the smallest sphere?

(a) 1.5 cm (b) 3 cm (c) 4 cm (d) 5 cm

16. Two solid spheres with radii of 1 cm and 6 cm are melted and recast into a hollow sphere with a thickness of 1 cm. Find the outer radius of the new hollow sphere and calculate its curved surface area.

17. A conical vessel with a radius of 6 cm contains some water. Several solid spheres, each with a radius of 1.5 cm, are completely submerged in the water. How many such spheres are required to raise the water level by 2 cm?

18. 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?

19. 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.

20. If two solid spheres with diameters of 1 cm and 6 cm are melted and recast into a hollow sphere with a thickness of 1 cm, then what is the outer surface area of the new hollow sphere?

21. 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.

22. Three solid copper spheres with radii of 3 cm, 4 cm, and 5 cm are melted and recast into a single solid larger sphere. Find the diameter of the larger sphere.

23. 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."

24. 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.

25. 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.

26. 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.

27. 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.

28. 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

29. Three solid copper spheres with diameters of 6 cm, 4 cm, and 10 cm are melted and recast into one large solid sphere. Find the diameter of the large sphere.

30. 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.