1. 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.

2. "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.)

3. 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.

4. 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?

5. 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.

6. 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.

7. 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.

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 to form a single larger solid sphere. Determine the diameter of the larger sphere.

10. 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.

11. 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.

12. The volume of a sphere is directly proportional to the cube of its radius. A lead sphere has a radius of 14 cm. Using the concept of proportion, prove that this sphere can be melted to form four spheres, each with a radius of 7 cm. (Assume that the volume remains constant during melting.)

13. We take a vertical cylindrical drum with radius 21 cm and height 21 cm, and a solid iron sphere with a diameter of 21 cm. We calculate the ratio of the volumes of the drum and the solid iron sphere (neglecting the thickness of the drum). Next, we completely fill the drum with water and then fully immerse the sphere into it. Now, determine the new depth of water in the drum after the sphere is removed.

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 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?

16. 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?

17. 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

18. 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.

19. 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.

20. 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.

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. 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.

23. 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.

24. If three metallic cubes with edges measuring 3 cm, 4 cm, and 5 cm are melted and recast into a new cube, the length of the edge of the new cube will be –

(a) 6 cm (b) 7 cm (c) 8 cm (d) 9 cm

25. 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

26. 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?

27. In front of Anowar's house, there is a solid iron pillar with a cylindrical lower part and a conical upper part. The diameter of the base is 20 cm, the height of the cylindrical part is 2.8 meters, and the height of the conical part is 42 cm. If the weight of 1 cubic centimeter of iron is 7.5 grams, then calculate and write the total weight of the iron pillar.

28. 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.

29. 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.