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

3. The volume of a solid sphere with radius \( 2r \) units.

(a) \(\cfrac{32πr^3}{3}\) cubic units (b) \(\cfrac{16πr^3}{3}\) cubic units (c) \(\cfrac{8πr^3}{3}\) cubic units (d) \(\cfrac{64πr^3}{3}\) cubic units

4. 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

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

6. 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

7. 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.

8. 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.

9. If a solid hemisphere has a radius of \(2r\) units, then its total surface area is ____ \(\pi r^2\) square units.

10. 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.

11. Two solid spheres with radii of 1 cm and 6 cm are melted to form a completely hollow sphere with a radius of 5 cm. Determine the volume and curved surface area of the new sphere.

12. The English translation is: "A solid gold sphere with a radius of 4.2 cm is melted and recast into a solid right circular cylinder with a radius of 2.8 cm. Determine the height (length) of the cylinder."

13. 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.

14. 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.

15. A solid sphere with a radius of \(r\) units is melted and recast into a solid right circular cone with a height of \(r\) units. Find the radius of the base of the cone.

(a) 2r units (b) 3r units (c) r unit (d) 4r units

16. 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.

17. "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.)

18. Find the total surface area of a solid hemisphere with radius \(x\) units.

19. A solid sphere with a radius of 3 cm is melted and recast into several small spherical balls, each with a diameter of 0.6 cm. How many such balls are formed?

(a) 1000 (b) 500 (c) 900 (d) 800

20. 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) \(9r\)

21. 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

22. A solid lead sphere with a diameter of 12 cm is melted to form three smaller spheres. If the diameters of the smaller spheres are in the ratio 3:4:5, what is the radius of the smallest sphere?

(a) 1.5 cm (b) 3 cm (c) 4 cm (d) 5 cm

23. 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

24. 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.

25. The numerical value of the volume and the total surface area of a solid hemisphere are equal. Find the radius of the hemisphere.

26. A hollow sphere made of lead sheet with a thickness of 1 cm has an outer radius of 6 cm. After melting the sphere, a solid cylindrical rod with a radius of 2 cm is formed. What is the length of the rod?

27. 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?

28. The radius of the base of a solid right circular cone is equal to the radius of a solid sphere. If the volume of the sphere is twice the volume of the cone, find the ratio of the height of the cone to the radius of its base.

29. 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. 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.