A solid metallic cube of edge 4 cm is melted and recast into solid cubes of edge 1 cm. If x is the surface area of the melted cube and y is the total surface area of the cubes recast, then what is `x:y` ?

Options

A. `2:1`
B. `1:2`
C. `1:4`
D. `4:1`

Answer

To solve the problem, we need to find the ratio \( x:y \) where \( x \) is the surface area of the original metallic cube and \( y \) is the total surface area of the smaller cubes after recasting. Let's break it down step by step.

### Step 1: Calculate the volume of the original cube
The edge length of the original cube is given as 4 cm. The volume \( V \) of a cube is calculated using the formula:
\[ V = \text{side}^3 \]
So, for the original cube:
\[ V = 4^3 = 64 \text{ cm}^3 \]

**Hint:** Remember that the volume of a cube is found by cubing the length of one of its edges.

### Step 2: Determine the number of smaller cubes
The smaller cubes have an edge length of 1 cm. The volume of one smaller cube is:
\[ V_{\text{small}} = 1^3 = 1 \text{ cm}^3 \]
To find the number of smaller cubes \( N \) that can be formed from the melted original cube, we use the total volume:
\[ N = \frac{\text{Volume of original cube}}{\text{Volume of one smaller cube}} = \frac{64}{1} = 64 \]

**Hint:** To find how many smaller cubes can be made, divide the volume of the original cube by the volume of one smaller cube.

### Step 3: Calculate the surface area of the original cube
The surface area \( S \) of a cube is calculated using the formula:
\[ S = 6 \times \text{side}^2 \]
For the original cube:
\[ S = 6 \times 4^2 = 6 \times 16 = 96 \text{ cm}^2 \]
Thus, \( x = 96 \text{ cm}^2 \).

**Hint:** The surface area of a cube is found by multiplying 6 by the square of the length of one edge.

### Step 4: Calculate the total surface area of the smaller cubes
Each smaller cube has a surface area of:
\[ S_{\text{small}} = 6 \times 1^2 = 6 \text{ cm}^2 \]
The total surface area \( y \) of all 64 smaller cubes is:
\[ y = N \times S_{\text{small}} = 64 \times 6 = 384 \text{ cm}^2 \]

**Hint:** To find the total surface area of multiple cubes, multiply the surface area of one cube by the number of cubes.

### Step 5: Find the ratio \( x:y \)
Now we can find the ratio \( x:y \):
\[ x:y = 96:384 \]
To simplify this ratio, we can divide both terms by 96:
\[ \frac{96}{96} : \frac{384}{96} = 1:4 \]

### Final Answer
Thus, the ratio \( x:y \) is:
\[ \boxed{1:4} \]