1. The volume of the largest solid cone that can be cut out from a solid hemisphere with a radius of r units.

(a) \(4\pi r^3\) cubic units. (b) \(43\pi r^3\) cubic units. (c) \(\cfrac{πr^3}{4} \)  cubic units (d) \(\cfrac{πr^3}{3}\) cubic 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. Determine the volume of the largest solid cone that can be cut from a solid hemisphere with a radius of \(r\) units.

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

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

6. 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.

7. 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.)

8. 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 \(a\)?

9. 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.

10. What is the ratio of the surface area of a cube to the curved surface area of the largest sphere that can be inscribed within it?

11. 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

12. 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.

13. 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.

14. 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

15. 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.

16. 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

17. 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

18. 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.

19. 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.

20. 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

21. 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

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 metallic sphere's outer surface is cut in such a way that the curved surface area of the remaining part becomes exactly half of the original sphere's curved surface area. Determine the ratio of the volume of the removed portion to that of the remaining portion.

24. 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.

25. 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

26. "The volume of a sphere is directly proportional to the cube of its radius. Three solid spheres with diameters of 1½, 2, and 2½ meters are melted to form a new solid sphere. Find the diameter of the new sphere. (Assume volume remains the same before and after melting.)

27. "The volume of a sphere varies directly with the cube of its radius, and the surface area of a sphere varies directly with the square of its radius. Prove that the square of the volume of a sphere will be directly proportional to the cube of its surface area."

28. A chord is drawn in a circle with a radius of 5 cm, at a distance of 3 cm from the center. What will be the length of that chord?

(a) 8 cm (b) 2 cm (c) 5 cm (d) 3 cm

29. If the ratio of the volume of a sphere with radius ???? to the volume of a cone with the same radius is 4:9, then the height of the cone will be..

(a) \(3r\) (b) \(4r\) (c) \(6r\) (d) \(9r\)

30. If the side length of a cube is equal to the radius of a sphere, then which one has the greater volume?