1. If a solid hemisphere and a solid cone have equal base areas and equal heights, then what is the ratio of their volumes? (a) 1:3 (b) 1:2 (c) 2:3 (d) 3:4

2. A cylinder and a hemisphere have the same radius and the same volume. By how many percent is the height of the hemisphere greater than the height of the cylinder? (a) 25% (b) 50% (c) 100% (d) 200%

3. What is the ratio of the volumes of a right circular cylinder and a cone having the same radius and height? (a) 1:3 (b) 3:2 (c) 3:1 (d) 2:3

4. If a hemisphere and a cone have equal base areas and equal heights, then what will be the ratio of their volumes? (a) 1:3 (b) 2:3 (c) 3:2 (d) 3:1

5. 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

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

7. 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

8. If a right circular cone and a sphere have the same radius, and the numerical value of the cone's volume is equal to the numerical value of the sphere's curved surface area, then what is the height of the cone? (a) 10 units (b) 11 units (c) 12 units (d) 13 units

9. If a sphere occupies the maximum space inside a cube, what will be the ratio of the volumes of the sphere and the cube (a) π:3 (b) π:2 (c) π:4 (d) π:6

10. 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) \(9r\)

11. If the radius of the base and the height of a cone are both doubled, then the volume of the cone becomes—? (a) 3 times (b) 4 times (c) 6 times (d) 8 times

12. The radius of the base of a right circular cone is equal to the radius of a sphere. The volume of the cone (in cubic centimeters) is equal to the curved surface area of the sphere (in square centimeters). What is the height of the cone? (a) 4 cm (b) 3 cm (c) 8 cm (d) 12 cm

13. A right circular cone and a right circular cylinder with the same base and the same height will have a volume ratio of 1 : 3. True / False

14. 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?

15. There is some water in a vertical cylindrical drum. A conical iron piece with a diameter of 2.8 decimeters and a height of 3 decimeters is completely submerged in the water, causing the water level to rise by 0.64 decimeters. What is the diameter of the drum?

16. 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?

17. A cylindrical vessel with a diameter of 12 cm is partially filled with water. A solid spherical metal ball is submerged into the vessel, causing the water level to rise by 1 cm. What is the diameter of the ball?

18. 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.

19. If a hemisphere and a cone have the same base area and the same height, find the ratio of their curved surface areas.

20. How many spheres of 6 cm diameter must be dropped into a partially water-filled right circular cone of 16 cm diameter so that the water level rises by 9 cm?

21. The length, width, and height of a rectangular block are 11 cm, 9 cm, and 6 cm respectively. How many coins of 3 cm diameter and ¼ cm thickness can be made from this block?

22. A solid sphere has a surface area of 616 square cm. It is melted to form 14 identical right circular cones, each with a height of 2 cm. Find the diameter of the base of each cone.

23. To make a hollow cylindrical pipe of uniform thickness, a solid iron sphere of radius 6 cm is melted. The pipe has a length of 18 cm and an outer radius of 10 cm. Find the thickness of the pipe.

24. A right circular cone and a cylinder have equal curved surface areas. If the height and radius of the cone are \(h\) and \(r\), and the height and radius of the cylinder are \(H\) and \(r\), then show that: \[ h^2 = (2H + r)(2H - r) \]

25. 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.

26. The lower part of an ice cream is conical in shape, and the upper part is hemispherical, both having the same base. If the height of the cone is 9 cm and the radius of the base is 2.5 cm, calculate the volume of the ice cream.

27. From a completely water-filled cubical tank, 75 buckets of equal size are taken out, after which \(\frac{2}{5}\) of the tank remains filled with water. If each edge of the tank is 1.5 meters long, how much water does each bucket hold in liters?

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

29. The shape of a one-end sharpened pencil is a combination of a cone and a ——.

30. A tank measuring 21 decimeters in length, 11 decimeters in width, and 6 decimeters in depth is half-filled with water. If 100 solid spheres, each with a diameter of 21 centimeters, are submerged into the tank, by how many decimeters will the water level rise?

31. একটি শঙ্কুর উচ্চতা 20 সেমি এবং তির্যক উচ্চতা 25 সেমি । শঙ্কুটির সমান আয়তনবিশিষ্ট একটি লম্ববৃত্তাকার চোঙের উচ্চতা 15 সেমি হলে চোঙটির ভূমির ব্যাস নির্ণয় করো । - translate in english

32. The shape of a one-end-sharpened pencil is a combination of a cylinder and a cone.

33. A hemisphere and a cone have the same base and height. Determine the ratio of their volumes and the ratio of their curved surface areas.

34. If water flows through a pipe with an internal diameter of 5.6 cm at a rate of 200 meters per minute, how long will it take to fill \(\frac{5}{8}\) of a tank that is 2.8 meters long, 2.2 meters wide, and 1.6 meters deep?

35. If a solid sphere and a solid right circular cylinder have the same radius and equal volume, then write the ratio of the cylinder's radius to its height.

36. Determine the ratio of the volumes of a solid right circular cylinder and a solid right circular cone, both having the same radius and height.

37. If a solid sphere is melted to form a solid right circular cylinder, then the _____ of the sphere and the cylinder will be equal.

38. 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.

39. The base length and width of a rectangular reservoir are 15 meters and 12 meters respectively. Water is filled into the reservoir from a nearby pond using a pump. If the pump can fill 36,000 liters of water per hour, then how long will it take for the pump to fill the reservoir up to a height of 7.2 decimeters? [Note: 1 liter = 1 cubic decimeter]

40. A solid object has its lower part in the shape of a hemisphere and its upper part in the shape of a right circular cone. If the surface areas of both parts are equal, determine the ratio of the radius to the height of the cone.

41. What is the ratio of the volumes of a solid right circular cylinder, a solid right circular cone, and a solid sphere, all having the same diameter and the same height?

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

43. A right circular cone and a hemisphere have equal base areas and volumes. Determine the ratio of their heights.

44. 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?

45. 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?

46. A conical vessel with a radius of 6 cm contains some water. Several solid spheres, each with a radius of 1.5 cm, are completely submerged in the water. How many such spheres are required to raise the water level by 2 cm?

47. A right circular cylinder and a cone have equal bases, and the ratio of their volumes is 3:2. Prove that the height of the cone is half the height of the cylinder.

48. A hemispherical vessel with an inner radius of 9 cm is completely filled with water. This water is poured into cylindrical bottles, each having a diameter of 3 cm and a height of 4 cm. How many such bottles are required to hold all the water?

49. If a cylinder and a sphere have the same volume with equal radius length, determine the ratio of the cylinder's diameter to its height.

50. There is some water in a vertical cylindrical drum. Three solid right circular cone-shaped iron pieces, each having a base diameter of 32 cm and height of 40 cm, are completely submerged in the water. As a result, the water level in the drum rises by 12.8 cm. Find the diameter of the drum.

51. A solid cylindrical iron rod of length 14 cm is melted to form 21 solid iron spheres. If the radius of each sphere is 8 cm, find the radius of the cross-section of the rod.

52. A roof measuring 13 meters in length and 11 meters in width had its drainage pipe closed during rainfall. After the rain, it was found that water had accumulated to a depth of 7 centimeters on the roof. The pipe through which the water drains has a diameter of 7 centimeters and discharges water in the form of a cylindrical stream at a rate of 200 meters in length per minute. Determine how long it will take for all the water to drain out once the pipe is opened.

53. A hollow iron cylinder of height 20 cm has an outer radius of 5 cm and an inner radius of 4 cm. This cylinder is melted and recast into a solid cone whose height is one-third of the original cylinder's height. Find the diameter of the base of the cone.

54. 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.