A cuboid has dimensions 8 cm × 10 cm × 12 cm. It is cut into smaller cubes of side 2 cm. What is the percentage increase in the total surface area after cutting the cuboid into these cubes?

Introduction / Context:
This question tests understanding of how surface area changes when a solid is cut into smaller pieces without any loss of material. A cuboid is divided into many small cubes, and although the total volume remains the same, the total surface area increases because new faces are created on the interior cuts. Calculating the percentage increase requires finding the initial total surface area and the final total surface area of all the small cubes combined.

Given Data / Assumptions:

• Original cuboid dimensions: 8 cm, 10 cm, and 12 cm.
• Small cubes each have side length 2 cm.
• All cuts are along planes parallel to the faces of the cuboid.
• No material is lost during cutting.
• We need the percentage increase in total surface area.

Concept / Approach:
Total surface area of a cuboid with length l, breadth b, and height h is TSA = 2(lb + bh + hl). Total surface area of a cube with side a is TSA = 6a^2. First compute the surface area of the original cuboid. Then determine how many 2 cm cubes can be cut from the cuboid by dividing each dimension by 2 and multiplying. The final total surface area is the number of cubes times surface area of one cube. The percentage increase is then calculated using the formula percentage increase = [(new area − old area) / old area] * 100.

Step-by-Step Solution:
Step 1: Original cuboid dimensions: l = 8 cm, b = 10 cm, h = 12 cm.Step 2: Original TSA = 2(lb + bh + hl) = 2(8*10 + 10*12 + 8*12) = 2(80 + 120 + 96) = 2 * 296 = 592 cm2.Step 3: Number of small cubes: each side is divided by 2. So along length 8/2 = 4 cubes, along breadth 10/2 = 5 cubes, along height 12/2 = 6 cubes. Total cubes = 4 * 5 * 6 = 120.Step 4: Surface area of one small cube with side 2 cm is 6 * 2^2 = 6 * 4 = 24 cm2.Step 5: Final total surface area of all cubes = 120 * 24 = 2880 cm2.Step 6: Increase in surface area = 2880 − 592 = 2288 cm2.Step 7: Percentage increase = (2288 / 592) * 100 ≈ 386.486 percent, which rounds to 386.5 percent.

Verification / Alternative check:
It is clear that cutting into smaller cubes will always increase the total surface area because many new faces become exposed. The original surface area of 592 cm2 is much smaller than the final total of 2880 cm2. The ratio 2880 / 592 ≈ 4.864, so the new area is about 4.864 times the old area. This corresponds to about 386.4 percent increase, which matches the detailed calculation and confirms the answer near 386.5 percent.

Why Other Options Are Wrong:

• 286.2 and 314.32 significantly underestimate the increase and do not match the ratio of new to old surface area.
• 250.64 is also too low and does not come from any correct ratio calculations in this scenario.
• 120.0 percent would mean the new area is only a bit more than double the old area, which is far from the actual factor of about 4.864.

Common Pitfalls:

• Forgetting to compute the number of cubes correctly by dividing each dimension, not by dividing volume directly.
• Using wrong formulas for surface area of cuboid or cube.
• Calculating percentage increase as new area divided by old area instead of the difference divided by old area.

Final Answer:
The percentage increase in total surface area is approximately 386.5 percent.