1. Determine the volume of the largest solid cone that can be cut from a solid hemisphere with a radius of \(r\) units.

2. From a solid hemisphere with radius \( r \) units, the volume of the largest solid cone that can be cut out will be _____ .

3. From a solid wooden cube with edge length 4.2 decimeters, determine the volume of the largest possible solid right circular cone that can be carved out with minimal wood wastage.

4. What is the volume of the largest cube that can be cut out from a sphere with radius \(4\sqrt{3}\) cm?

(a) 512 cubic cm (b) 64 cubic cm (c) 216 cubic cm (d) 729 cubic cm

5. Determine the volume of the largest possible solid right circular cone that can be obtained from a solid wooden cube with an edge length of 4.2 decimeters, ensuring minimal wood wastage.

6. From a solid wooden cube with an edge length of 4.2 decimeters, find the volume of the largest solid right circular cone that can be carved from it with minimum wood wastage.

7. From a solid hemisphere with a radius of \(r\) units, the largest possible solid cone is carved out. Find its volume.

(a) \(4πr^3\) cubic units (b) \(3πr^3\) cubic units (c) \(\frac{πr^3}{4}\) cubic units (d) \(\frac{πr^3}{3}\) cubic units

8. A solid vertical cylindrical cone has both its diameter and height equal to 21 cm. Find the volume of the largest possible sphere that can be obtained from this cone. Also, determine the ratio of the volumes of the cone and the sphere.

9. Determine the volume of the largest possible solid right circular cylinder that can be obtained from a solid wooden cube with an edge length of 4.2 decimeters, ensuring minimal wood wastage.

10. From a metallic sphere of radius \(4\sqrt{2}\) cm, find the volume of the largest cube that can be cut out.

11. What is the ratio of the volumes of a solid right circular cone and a solid sphere that can be carved with minimal wood wastage from two solid cubes, each having edge length \(a\)?

12. From a wooden cube with an edge of 4.2 units, what will be the volume of the largest right circular cone that can be obtained with minimum wood wastage?

(a) 19.808 cubic units (b) 19.202 cubic units (c) 19.404 cubic units (d) 19.303 cubic units

13. 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.

14. What is the volume of the largest right circular cone that can be obtained from a cube with an edge length of 28 cm?

15. 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

16. 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?

17. If the largest possible solid sphere is cut from a solid cube with an edge length of \(x\) units, then the diameter of the sphere will be – ?

(a) \(x\) unit (b) \(\cfrac{x}{2}\) unit (c) \(2x\) unit (d) \(4x\) unit

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. From a solid cube with side length \(x\) units, the largest possible solid sphere is carved out. Find the diameter of the sphere.

(a) \(x\) units (b) \(x\) units (c) \(\cfrac{x}{2}\) unitsts (d) \(4x\) units

20. Two identical solid hemispheres, each with a base radius of \(r\) units, are joined together along their flat surfaces. The total surface area of the resulting solid body will be \(6πr^3\) square units.

21. 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

22. How many solid cones of the same radius and height can be made by melting a solid cylinder?

23. A solid hemisphere and a solid cone have the same base radius and height. What is the ratio of their volumes?

(a) 3:2 (b) 1:3 (c) 2:3 (d) None of the above

24. 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.

25. 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.)

26. 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?

27. 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.

28. 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?

29. If a solid hemisphere and a solid cone have the same base radius and height, what is the ratio of their volumes?

(a) 3:2 (b) 1:3 (c) 2:3 (d) None of the above

30. Determine the volume ratio of a solid cone, a solid hemisphere, and a solid cylinder with equal base diameters and equal heights.