1. 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.

2. 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?

3. 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.

4. 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.

5. Two solid spheres with radii 1 cm and 6 cm are melted and recast into a hollow sphere of thickness 1 cm. Calculate and write the curved surface area of the new hollow sphere.

6. 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.

7. 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?

8. 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.

9. Three solid copper spheres with radii of 3 cm, 4 cm, and 5 cm are melted and used to form a single larger solid sphere. Find the radius of the larger sphere.

10. 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.

11. 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.

12. 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

13. 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.

14. Three solid copper spheres with radii of 3 cm, 4 cm, and 5 cm are melted to form one larger solid sphere. Find the radius of the larger sphere.

15. Three solid copper spheres with radii of 3 cm, 4 cm, and 5 cm are melted to form a single larger solid sphere. Determine the diameter of the larger sphere.

16. Three solid copper spheres with radii of 3 cm, 4 cm, and 5 cm are melted to form one solid large sphere. Calculate the radius of the large sphere.

17. 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.

18. "The volume of a sphere is directly proportional to the cube of its radius. Three solid spheres with diameters of 1½, 2, and 2½ meters are melted to form a new solid sphere. Find the diameter of the new sphere. (Assume volume remains the same before and after melting.)

19. 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.

20. The radii of two circles are \(r_1\) and \(r_2\) units, respectively. The distance between their centers is \(d\) units. If the circles are externally tangent, which of the following will be correct?

(a) \(r_1-r_2=d\) (b) \(r_1+r_2\gt d\) (c) \(r_1+r_2=d\) (d) \(r_1-r_2\lt d\)

21. Three solid cubes have edge lengths of 3 cm, 4 cm, and 5 cm respectively. These cubes are melted and recast into a new solid cube. Write the length of an edge of the new cube.

22. If the ratio of the total surface areas of two solid spheres is 1:4, what will be the ratio of their volumes? Calculate and write it down.

23. Two identical solid hemispheres, each with a base radius of \(r\) units, are joined together along their flat surfaces. The total surface area of the resulting solid body will be \(6πr^3\) square units.

24. Three solid copper spheres with radii of 3 cm, 4 cm, and 5 cm are melted to form one larger solid sphere. Find the diameter of the larger sphere.