1. If the numerical value of the volume and the surface area of a sphere are equal, what is the numerical value of the radius of the sphere? (a) 3 (b) 5 (c) 7 (d) 4

2. How much brass sheet is required to make a hemispherical brass bowl with a diameter of 1.4 decimeters? (a) 3.08 cubic decimeters. (b) 718.66 cubic decimeters. (c) 3.08 square decimeters (d) 30.8 square decimeters

3. The diameter of one sphere is twice the diameter of another sphere. The curved surface area of the first sphere is equal to the volume of the second sphere. What is the radius of the first sphere? (a) 21 (b) 22 (c) 23 (d) 24

4. If the total surface area of a sphere is \( A \) and its volume is \( V \), what is the relationship between them ? (a) \(A^3=36πV^2\) (b) \(A^3=36V^3 \) (c) \(A^2=4V^2\) (d) \(A=3V\)

5. If the total surface area of a solid sphere is 462 square centimeters, what is its diameter? (a) 7 cm (b) 14 cm (c) 21 cm (d) 3.5 cm

6. If the numerical values of the curved surface area and the total surface area of a sphere are equal, what is its radius? (a) 3 units (b) 4 units (c) 5 units (d) 6 units

7. The outer radius and inner radius of a hollow sphere made of iron are 21 cm and 7 cm respectively. If the weight of iron is 30 grams per cubic centimeter, what is the weight of the sphere? (a) 2112.12 kg (b) 2212.21 kg (c) 1121.12 kg (d) 1122.21 kg

8. How many solid spheres of radius 1 cm can be made by melting a solid iron ball of radius 20 cm? (a) 20 (b) 400 (c) 800 (d) 8000

9. How many solid spheres of radius 3 cm can be made by melting a solid cone of radius 2 cm and height 45 cm? (a) 5 (b) 12 (c) 14 (d) 7

10. A cylindrical rod of height 64 cm is melted to form 12 solid spheres of equal radius. What is the radius of each sphere? (a) 2 cm (b) 4 cm (c) 8 cm (d) 16 cm

11. If the radius of a sphere is increased by 2 cm, its curved surface area increases by 352 square centimeters. What was the original radius of the sphere? (a) 5 cm (b) 6 cm (c) 7 cm (d) 5.6 cm

12. 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

13. The volume ratio of two solid spheres is 4:27. The ratio of their curved surface areas is- (a) 4:9 (b) 2:3 (c) 16:35 (d) 8:27

14. If the radius of a sphere is increased by 3%, by what percentage will its surface area increase? (a) 6.09% (b) 7% (c) 5.06% (d) 9%

15. 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

16. If the volume ratio of two solid spheres is 1:8, the ratio of their curved surface areas will be - (a) 1:2 (b) 2:1 (c) 1:4 (d) 1:16

17. What is the surface area of a sphere with a radius of 4 cm? (a) \(64\pi\)square cm (b) \(69\pi\)square cm (c) \(32\pi\)square cm (d) \(35\pi\)square cm

18. 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

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. The volume ratio of a sphere and the circumscribed cube of the sphere is (a) π:2 (b) π:3 (c) π:4 (d) π:6

21. If the radius of a sphere is decreased by 10%, by what percentage will the volume decrease? (a) 10% (b) 25% (c) 27.1% (d) 29.7%

22. A sphere and a right circular cone have the same radius \(r\) and equal volume. What is the height of the cone? (a) \(\cfrac{r}{3}\) (b) \(\cfrac{r}{4}\) (c) \(\cfrac{3r}{4}\) (d) \(\cfrac{4r}{3}\)

23. If \(A\) is the total surface area and \(V\) is the volume of a sphere, which of the following is correct? (a) \(A^3=36πV^2\) (b) \(A^2=4V^2\) (c) \(A^3=36V^2\) (d) \(A^3=36V^3\)

24. If \(S\) is the curved surface area and \(V\) is the volume of a sphere, the value of \(\cfrac{S^3}{V^2}\) is – (a) 4π (b) 12π (c) 36π (d) 72π

25. The volume of a solid sphere with a radius of \(2r\) units is: ____? (a) \(\cfrac{32πr^3}{3}\) cubic unit (b) \(\cfrac{16πr^3}{3}\) cubic unit (c) \(\cfrac{8πr^3}{3}\) cubic unit (d) \(\cfrac{64πr^3}{3}\) cubic unit

26. After a spherical ice ball starts melting, it is observed that its curved surface area has reduced by 75% after 15 minutes. The percentage of ice that has melted during this time is – (a) 75 (b) 87.5 (c) 50 (d) 92.5

27. If the volumes of two solid spheres are in the ratio 27:8, what will be the ratio of their curved surface areas? (a) 1:2 (b) 9:4 (c) 1:8 (d) 1:16

28. 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) \((r_1^3+r_2^3)^{\cfrac{1}{3}}\)

29. The volume of a sphere with radius \(r\) cm will be: \[ \text{Volume} = \frac{4}{3} \pi r^3 \text{ cubic centimeters} \] (a) (b) (c) (d)

30. The diameter of one sphere is twice the diameter of another sphere. The curved surface area of the first sphere is equal to the volume of the second sphere. What is the radius of the first sphere? (a) 21 (b) 22 (c) 23 (d) 24

31. The numerical value of a sphere’s volume and surface area are equal. What is the numerical value of the sphere’s radius? (a) 2 (b) 3 (c) 4 (d) 5

32. If the total surface area of a hemisphere is \(36\pi\) square centimeters, then its radius will be 3 cm. True / False

33. If the radius of a sphere is doubled, then its volume will be twice the volume of the original sphere. True / False

34. If the numerical value of the curved surface area and the volume of a sphere are equal, then the radius will be 3 units. True / False

35. If the surface area of a hemisphere is \( 27\pi \) square cm, then its radius will be 3 cm. True / False

36. Two solid spheres with radii 1 cm and 6 cm are melted to form a hollow sphere with an outer radius of 9 cm. Find the inner radius of the hollow sphere.

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

38. If the surface area of a sphere is \(A\) and its volume is \(V\), then find the value of \(\cfrac{A^3}{V^2}\).

39. The number of surfaces of a solid hemisphere is ——.

40. The ratio of the total surface areas of a solid sphere and a solid hemisphere of the same radius is 2:1.

41. If the radius of a sphere is doubled, by what percentage does its curved surface area increase?

42. The diameter of one sphere is twice the diameter of another sphere. If the numerical value of the surface area of the larger sphere is equal to the numerical value of the volume of the smaller sphere, then what is the radius of the smaller sphere?

43. If the curved surface area of a solid hemisphere is \( S \) and its volume is \( V \), find the value of \( \cfrac{S^3}{V^2} \).

44. If the radius of a sphere is increased by 50%, by what percentage will its curved surface area increase?

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

46. If the volume of a solid hemisphere is \(144π\) cubic centimeters, what is the diameter of the sphere?

47. If a solid sphere with a diameter of 21 cm is completely submerged in water, how many liters of water will be displaced?

48. Translate the following: A solid iron sphere of radius 8 cm is melted and recast into solid spherical balls, each of radius 1 cm. Determine how many such balls can be made.

49. If the diameter of a solid sphere is doubled, its volume becomes four times greater.

50. If the diameter of a solid sphere is tripled, its surface area will be _____ times larger.

51. If the curved surface area of a solid sphere and three times its volume are numerically equal, then the diameter of the sphere is _____ units.

52. How many solid spheres of radius 5 mm can be made by melting a cylindrical iron rod that is 1 meter long and has a diameter of 4 cm?

53. If the curved surface area of a solid sphere is \(s\) and its volume is \(v\), then what is the value of \(\frac{s^3}{v^2}\)?

54. The curved surface area of a sphere decreases from \(16\pi\) square centimeters to \(4\pi\) square centimeters. By what percentage will the volume of the sphere decrease?

55. If the ratio of the curved surface areas of two spheres is 1 : 4, find the ratio of their volumes.

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

57. 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.

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

59. If the numerical values of the volume and surface area of a solid sphere are equal, then what is the numerical value of its diameter?

60. The curved surface area of a sphere is 5544 square cm. Find the volume of the sphere.

61. If the curved surface area of a sphere is \(S\) square units and its volume is \(V\) cubic units, then determine the relationship between \(S\) and \(V\).

62. The total surface area of a solid hemisphere and the curved surface area of a solid sphere are equal. What is the ratio of the radius of the hemisphere to the radius of the sphere?

63. Three solid copper spheres with diameters of 6 cm, 4 cm, and 10 cm are melted and recast into one large solid sphere. Find the diameter of the large sphere.