1. 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.

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. 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.

4. 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.

5. 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.

6. 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.

7. 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.

8. 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.

9. 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.

10. From a cuboid measuring 16 cm × 4 cm × 2 cm, several cubes with edge length 2 cm are cut out. What is the ratio of the total surface area of the original cuboid to the total surface area of all the cubes?

(a) 11:48 (b) 12:50 (c) 13:36 (d) 14:28

11. 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?

12. 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?

13. 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.

14. 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.

15. 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.

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. 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.

18. - sin 280 sec 210 - 1 The dimensions of a cuboid are 12 cm, 6 cm, and 3 cm. What will be the length of each edge of a cube having the same volume as that cuboid?

19. Several coins, each with a diameter of 1.5 cm and thickness of 0.2 cm, are melted to form a solid vertical cylindrical rod with a height of 10 cm and diameter of 4.5 cm. Determine the number of coins used.

20. 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

21. "The differences between the semi-perimeter of a triangle and the lengths of its sides are respectively 4 cm, 7 cm, and 5 cm. The area of the triangle is –"

(a) \(20\sqrt{7}\) square cm (b) \(10\sqrt{14}\) square cm (c) \(20\sqrt{14}\) square cm (d) 140 square cm

22. 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.

23. A rectangular container has a base in the shape of a rectangle with a length of 60 cm and a width of 45 cm. The container has a height of 20 cm and is half-filled with water. Determine the side length of a metallic cube that, when placed in the container, will cause the water level to reach the brim.

24. If the total length of the edges of a solid cube is 36 cm, then its volume will be –?

(a) 9 cubic cm (b) 54 cubic cm (c) 27 cubic cm (d) 36 cubic cm

25. The radii of two circles are 5 cm and 3 cm, respectively. If the circles touch each other externally, then the distance between their centers will be –

(a) 2 cm (b) 3 cm (c) 8 cm (d) 1.5 cm

26. If the length of a diagonal of a cube is \(4\sqrt{3}\) cm, calculate and write the total surface area of the cube.

27. The dimensions of a cuboid are 12 cm, 6 cm, and 3 cm respectively. Calculate the length of each edge of a cube whose volume is equal to that of the cuboid.