1. How many solid spheres of radius 5 mm can be made by melting a cylindrical iron rod that is 1 meter long and has a diameter of 4 cm?
2. How many solid iron balls of radius 1 cm can be made by melting one solid iron sphere of radius 4 cm?
3. How many solid spheres of radius 3 cm can be made by melting a solid cone of radius 2 cm and height 45 cm?
(a) 5 (b) 12 (c) 14 (d) 7
4. Let’s calculate and see how many solid iron balls of 1 cm diameter can be obtained by melting a solid iron sphere of radius 8 cm.
5. A right circular cone has a base radius of 20 cm and a height of 10 cm. If it is melted down, how many solid spheres of radius 2 cm can be made from it?
(a) 200 (b) 100 (c) 250 (d) 125
6. If a solid iron sphere has a radius of 12 cm, how many cone-shaped objects of 2 cm radius and 3 cm height can be formed by melting the sphere?
7. 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.
8. 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.
9. 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?
10. If a solid iron sphere with a radius of 4 cm is melted, how many solid bullets with a radius of 1 cm can be made?
11. The diameter of a solid lead sphere is 14 cm. If this sphere is melted, calculate how many solid spheres can be made from it, each having a radius of 3.5 cm.
12. How many solid cones of the same radius and height can be made by melting a solid cylinder?
13. 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?
14. A hollow cube is made using copper sheets that are 1.5 cm thick, with an external edge length of 20 cm. After melting the cube, how many solid cuboids measuring 7 cm × 3 cm × 1 cm can be made from it?
(a) 200 (b) 220 (c) 100 (d) 147
15. 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?
16. 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.
17. 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?
18. 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.
19. The radii of two solid iron spheres are \(r_1\) and \(r_2\) respectively. The spheres are melted and recast into a single solid sphere. What will be the radius of the resulting sphere?
(a) \((r_1^3+r_2^3)\) (b) \((r_1^3+r_2^3)^3\) (c) \((r_1+r_2)^3\) (d) \((r_1^3+r_2^3)^{\cfrac{1}{3}}\)
20. A hollow sphere made of lead sheet with a thickness of 1 cm has an outer radius of 6 cm. After melting the sphere, a solid cylindrical rod with a radius of 2 cm is formed. What is the length of the rod?
21. 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.
22. 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.)
23. A solid cylindrical iron rod of length 14 cm is melted to form 21 solid iron spheres. If the radius of each sphere is 8 cm, find the radius of the cross-section of the rod.
24. 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?
25. A solid cylindrical iron rod of length 14 cm is melted to form two solid iron spheres. If the radius of each sphere is 4 cm, what was the radius of the cross-section 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 the sphere is melted to form a solid cylindrical rod with a radius of 2 cm, what will be the length of the rod?
27. "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."
28. 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.
29. A table showing how many of the 150 athletes completed the 100-meter hurdles race within different time intervals (in seconds) is given below. ```html
30. 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).
