1. If the diameter of a right circular cylinder is 3 cm and its height is 4 cm, then the longest rod that can be placed inside the cylinder will have a length of ___________ cm.

2. A one-end open cylindrical pipe is 12 meters long and has a radius of 4.5 meters. What is the length of the longest rod that can be placed inside the cylinder?

3. 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.

4. The length, width, and height of a room are 5 m, 4 m, and 3 m respectively. What is the maximum length of a rod that can be placed inside the room?

(a) \(5\sqrt2\) meters (b) \(7\) meters (c) \(8\sqrt2\) meters (d) \(10\sqrt2\) meters

5. If the internal length, width, and height of a rectangular box are 6 meters, 3 meters, and 2 meters respectively, what is the length of the longest rod that can be placed inside the box?

6. The length, width, and height of a cuboidal room are respectively \(a\), \(b\), and \(c\) units, and given that \(a + b + c = 25\) and \(ab + bc + ca = 240.5\), what is the length of the longest rod that can be placed inside the room?

7. If the base radius of a right circular cone is 3 cm and the height is 4 cm, then the lateral surface area of the cone will be.

(a) \(10\pi \, cm^2\) (b) \(15\pi \, cm^2\) (c) \(12\pi \, cm^2\) (d) \(18\pi \, cm^2\)

8. 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?

9. 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?

10. 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.

11. 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.

12. If the length, breadth, and height of a rectangular room are 5 m, 4 m, and 3 m respectively, calculate and write the length of the longest rod that can be placed inside the room.

13. The radius of a right circular cylinder is \(r\) units and its height is \(2r\) units. Find the diameter of the largest sphere that can be placed inside the cylinder.

(a) r unit (b) 2r units (c) \(\frac{r}{2}\) units (d) 4r units

14. If a right circular cone has a height of 12 cm and a volume of \(100\pi\) cubic cm, then what is the radius of the cone?

15. A vertical circular cylinder has a height that is twice its radius. If the height were six times the radius, then the volume of the cylinder would be 539 cubic decimeters more. What is the height of the cylinder?

16. A rectangular wooden box has the sum of its dimensions equal to 10 cm. The longest rod that can be placed inside it has a length of \(\sqrt{38}\) cm. Find the total surface area of the box.

17. A rod of length \(6\sqrt{3}\) cm can be placed inside a cube along its longest diagonal. What is the volume of the cube?

18. 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?

19. The maximum length of a rod that can be placed inside a cube is \(6\sqrt3\) cm. What is the volume of the cube?

20. In a cuboid-shaped room, the length, breadth, and height are respectively \(a\), \(b\), and \(c\) units. Given \(a + b + c = 25\) and \(ab + bc + ca = 240.5\), calculate the length of the longest rod that can be placed inside the room.

21. If the height of a right circular cylinder is 14 cm and the curved surface area is 264 square cm, then what is the volume of the cylinder?

22. If a right circular cone has a vertex angle of 60° and a base radius of 14 cm, then what is the curved surface area of the cone?

(a) 1680 square cm (b) 1232 square cm (c) 1080 square cm (d) 1442 square cm

23. A rod of length \( 4\sqrt{3} \) cm is placed inside a cube along its longest diagonal. What is the volume of the cube?

24. ABC and POR are two similar triangles. If BC = 5 cm, QR = 4 cm, and the height AD = 3 cm, then what is the length of the height PE?

(a) 4.2 cm (b) 1.25 cm (c) 5.4 cm (d) 2.4 cm

25. If a right circular cone has a height of \(12\) cm and a volume of \(1024\pi\) cubic cm, what is the slant height of the cone?

(a) 20 cm (b) 21 cm (c) 22 cm (d) 23 cm

26. 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.

27. A hollow metallic sphere has an inner radius of 3 cm and an outer radius of 5 cm. It is melted to form a solid right circular cylinder of height \(\frac{8}{3}\) cm. Find the diameter of the base of the cylinder.

28. If the ratio of the heights of two right circular cylinders is 1:2 and the ratio of their base circumferences is 3:4, then the ratio of their volumes will be 9:32.

29. A right circular cylinder has a height-to-base-radius ratio of 3:1. If the volume of the cylinder is \(1029\pi\) cubic cm, find the total surface area of the cylinder.

30. A sphere and a right circular cylinder both have a radius of 3 cm. If their volumes are equal, what is the height of the cylinder?