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

2. 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

3. 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

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. 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.

6. 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.

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

8. 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

9. 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.

10. 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

11. 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.

12. 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

13. 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 \(a\)?

14. 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.

15. 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.

16. If the radius of a hemisphere is \(2r\) units, then the total surface area will be _____.

17. 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.

18. The radii of two solid iron spheres are \(r_1\) and \(r_2\) respectively. The spheres are melted and recast into a single solid sphere. What will be the radius of the resulting sphere?

(a) \((r_1^3+r_2^3)\) (b) \((r_1^3+r_2^3)^3\) (c) \((r_1+r_2)^3\) (d) \((r_1^3+r_2^3)^{\cfrac{1}{3}}\)

19. Two identical circles, each with radius \(r\), intersect in such a way that each circle passes through the center of the other. The centers of the circles are labeled A and B, and they intersect at points P and Q. The area of triangle \(\triangle APB\) will be:

(a) \(\cfrac{\sqrt3}{4}r^2\) (b) \(\cfrac{\sqrt3}{2}r^2\) (c) \(\cfrac{\sqrt3}{3}r^2\) (d) \(\sqrt3 r^2\)

20. The volume of a sphere with radius \(\cfrac{r}{2}\) units will be \(\cfrac{4}{3}\pi\left(\cfrac{r}{2}\right)^3 = \cfrac{\pi r^3}{6}\).

(a) \(\cfrac{1}{6}\pi r^3\) cubic units (b) \(\cfrac{4}{3}\pi r^3\) cubic units (c) \(\cfrac{2}{3}\pi r^3\)cubic units (d) \(\cfrac{1}{3}\pi r^3\)cubic units

21. 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?

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 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.

24. If the radius of a solid hemisphere is \(3r\) units, then its total surface area is _____.

25. If a solid cylinder has a height of 7 cm and a volume of \(28\pi\) cubic cm, then the curved surface area of the cylinder will be -.

(a) \(14\pi\) square cm (b) \(28\pi\) square cm (c) \(56\pi\) square cm (d) \(112\pi\) square cm

26. The curved surface area of a right circular cone is \(\sqrt{5}\) times its base area. The ratio of its height to radius will be _____.

27. A solid rod has a length of \( h \) meters and a diameter of \( r \) meters. It is melted to form 6 spheres, each with a radius of \( r \) meters. Determine the relationship between \( h \) and \( r \).

28. 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?

29. If the radius of a solid hemisphere is 2r units, then the total surface area is _______ \( πr^2 \) square units.

30. 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.