1. 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?

2. 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.

3. 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.

4. 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.

5. To make a hollow cylindrical pipe of uniform thickness, a solid iron sphere of radius 6 cm is melted. The pipe has a length of 18 cm and an outer radius of 10 cm. Find the thickness of the pipe.

6. A hollow cylinder with a height of 13 cm has an outer radius of 6.75 cm and an inner radius of 5.25 cm. It is melted to form a solid cylinder with a height of 6.5 cm. Find the radius of the solid cylinder.

7. 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.

8. 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.

9. 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

10. 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?

11. A right circular drum has a radius of 21 cm and a height of 2i. A solid iron sphere has a diameter of 21 cm. Determine the ratio of the volumes of the drum and the solid iron sphere (neglecting the thickness of the drum). Now, if the drum is completely filled with water and the sphere is fully submerged and then removed, determine the new depth of the water in the drum.

12. 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.

13. 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.

14. 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.

15. 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.)

16. 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.

17. A vertical hollow cylindrical pipe made of iron has both ends open. Its height is 35 cm, the outer radius is 12 cm, and the inner radius is 10 cm. What is the volume of iron in the pipe in cubic centimeters?

18. A circle with a radius of 10 cm has two parallel chords of lengths 4 cm and 6 cm. What is the distance between the two chords?

(a) \(\sqrt7(\sqrt3-\sqrt{12})\) cm (b) \(\sqrt3(\sqrt7+\sqrt{12})\) cm (c) \(\sqrt{13}(\sqrt{12}-\sqrt{7})\) cm (d) \(\sqrt{7}(\sqrt{13}+\sqrt{12})\) cm

19. 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

20. From point A, which is located 26 cm away from the center O of a circle, a tangent is drawn to the circle with a length of 10 cm. Find the radius of the circle.

21. A vertical cylindrical drum with a radius of 21 cm and height of 21 cm is completely filled with water. A solid sphere with a radius of 21 cm is taken and fully submerged into the drum, then removed. As a result, what is the new depth of water in the drum?

22. A hollow vertical cylindrical iron pipe has an outer radius of 5 cm and an inner radius of 4 cm. If the total surface area of the pipe is 1188 square cm, what is the length of the pipe?

23. A right circular cylinder made of iron sheet is open at both ends. Its height is 42 cm, thickness is 1 cm, and its outer radius is 10 cm. Find the amount of iron used to make the cylinder.

24. 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?

25. 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?

26. 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?

27. 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?

28. Point P is an external point to a circle with center O. The distance from point P to the center of the circle is 26 cm, and the length of the tangent drawn from point P to the circle is 10 cm. The radius of the circle is ____ cm.

29. A cylindrical vessel with a radius of 12 cm contains some water. If 60 solid iron cones, each with a base diameter of 6 cm and a height of 4 cm, are completely submerged in the water, how much will the water level rise?

30. A cylindrical drum has a radius of \(r\) cm and a height of \(h\) cm. If the drum is half-filled with water, the volume of the water will be \(\cfrac{1}{2}\pi r^2 h\) cubic cm.