A big cube of side 8 cm is formed by rearranging together 64 small but identical cubes each of side 2 cm . Further , if the corner cubes in the topmost layer of the big cube are removed , what is the change in total surface area of the big cube ?

Options

A. `16 cm^(2)`, decreases
B. `48 cm^(2)`, decreases
C. `32 cm^(2)`, decreases
D. remains the same as previously

Answer

To solve the problem, we need to calculate the change in the total surface area of the big cube after removing the corner cubes from the topmost layer. Here’s a step-by-step solution:

### Step 1: Calculate the surface area of the big cube before removing the corner cubes.
The formula for the surface area \( A \) of a cube with side length \( s \) is given by:
\[ A = 6s^2 \]
For the big cube with a side length of 8 cm:
\[ A = 6 \times (8 \, \text{cm})^2 = 6 \times 64 \, \text{cm}^2 = 384 \, \text{cm}^2 \]

### Step 2: Determine the number of corner cubes removed.
In the topmost layer of the big cube, there are 4 corner cubes.

### Step 3: Calculate the surface area contribution of the removed corner cubes.
Each small cube has a side length of 2 cm. The surface area of one small cube is:
\[ A_{\text{small}} = 6 \times (2 \, \text{cm})^2 = 6 \times 4 \, \text{cm}^2 = 24 \, \text{cm}^2 \]
When a corner cube is removed, it exposes 3 new faces. Therefore, the change in surface area for one removed corner cube is:
\[ \text{Change for one cube} = 3 \times (2 \, \text{cm})^2 = 3 \times 4 \, \text{cm}^2 = 12 \, \text{cm}^2 \]
For 4 corner cubes, the total change in surface area is:
\[ \text{Total change} = 4 \times 12 \, \text{cm}^2 = 48 \, \text{cm}^2 \]

### Step 4: Calculate the new total surface area after removing the corner cubes.
The new total surface area after removing the corner cubes is:
\[ \text{New Surface Area} = \text{Original Surface Area} - \text{Area of removed cubes} + \text{Exposed area} \]
Since we are removing the area of the cubes but adding the exposed area, the total change in surface area is:
\[ \text{New Surface Area} = 384 \, \text{cm}^2 - 0 + 48 \, \text{cm}^2 = 384 \, \text{cm}^2 \]
Thus, the total surface area remains the same.

### Step 5: Conclusion
The change in total surface area of the big cube after removing the corner cubes is **0 cm²**.

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