1. If each edge of a cube is reduced by 50%, calculate and write the ratio of the volume of the original cube to the volume of the altered cube.

2. Each edge length of a cube is increased by 25%. Determine the ratio of the volume of the original cube to the volume of the modified cube.

3. If each edge of a cube is doubled in length, then the volume of the new cube will be eight times that of the original cube.

4. The diameter of the cross-section of a wire is reduced by 50%. To keep the volume unchanged, by what percentage must the length of the wire be increased?

5. 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

6. If each edge of a cube is decreased by 10%, what is the percentage decrease in its total volume?

(a) 36.2% (b) 27.1% (c) 30.5% (d) 20.6%

7. A sphere is perfectly enclosed within a cube. What is the ratio of the volume of the sphere to the volume of the cube?

8. Two cubes with equal edge lengths are joined side by side to form a cuboid. If the volume of the cuboid is 1024 cubic centimeters, what is the length of each edge of the cubes?

9. If the length of each edge of a cube is doubled, the volume of the new cube will be 4 times that of the original cube.

10. The dimensions of a cuboid are 12 cm, 6 cm, and 3 cm respectively. Calculate the length of each edge of a cube whose volume is equal to that of the cuboid.

11. 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 \(a\)?

12. The volume of a cube is equal to the area of one of its faces. What is the length of its edge?

(a) 4 units (b) 5 units (c) 6 units (d) 8 units

13. The cube whose total edge length is 60 cm has a volume of —

(a) 110 cubic cm (b) 120 cubic cm (c) 125 cubic cm (d) 130 cubic cm

14. If each edge of a cube is increased by 20%, by what percentage will the total surface area increase?

(a) 40% (b) 42% (c) 44% (d) 46%

15. Let the length of each edge of the cube be \(a\) cm \(\therefore\) The total surface area of the cube = \(6a^2\) square cm If the edge length is increased by 20%, the new edge length = \(a + a \times \cfrac{20}{100}\) cm \(= a + \cfrac{a}{5} = \cfrac{6a}{5}\) cm \(\therefore\) New total surface area of the cube = \(6\left(\cfrac{6a}{5}\right)^2\) square cm = \(\cfrac{216a^2}{25}\) square cm \(\therefore\) Percentage increase in surface area \(= \cfrac{\cfrac{216a^2}{25} - 6a^2}{6a^2} \times 100\%\) \(= \cfrac{(216a^2 - 150a^2) \times 100}{25 \times 6a^2}\%\) \(= \cfrac{4 \times 66a^2}{6a^2}\%\) \(= 44\%\)

16. If the volume of a cube is \(V\) cubic centimeters, the total surface area is \(S\) square centimeters, and the length of the diagonal is \(d\) centimeters, then prove that \(Sd = 6\sqrt{3}V\).

17. If the ratio of the volumes of two cubes is 8:27, then what will be the ratio of their total surface areas?

18. 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?

19. If the length of each edge of a cube is increased by 50%, by what percentage will its volume increase?

20. 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.)

21. The sum of the edges of a cube is 60 cm. What is the volume of the cube?

22. If the ratio of the volumes of two cubes is 1:27, then the ratio of their total surface areas will be —

(a) 1:3 (b) 1:8 (c) 1:9 (d) 1:18

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

24. If each edge of a cube is increased by 10%, by what percentage does the total surface area increase?

(a) 30.2% (b) 39.6% (c) 33.1% (d) 38.3%

25. If the length of each edge of a cube is doubled, the volume of the new cube will be 4 times the volume of the original cube.

26. The radius of a right circular cone is reduced by 50% and its height is increased by 50%. By what percentage does the volume of the cone change?

27. The numerical value of the volume of a cube is equal to the numerical value of the sum of its edges. The total surface area of the cube is - -

(a) 12 square unit (b) 36 square unit (c) 72 square unit (d) 144 square unit

28. If the length of each edge of a cube is doubled, then by what percentage will the total surface area of the cube increase?

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

30. If the area of one face of a cube is 4 times the area of a face of another cube, then how many times is the volume of the first cube compared to the volume of the second cube?