A metallic cube of edge `2.5` cm is melted and recasted into the form of a cuboid of base `1.25` cm `xx 0.25` cm. Find the increase in the surface area.

Options

A. `123.325" cm"^(2)`
B. `150.625" cm"^(2)`
C. `113.125" cm"^(2)`
D. `37.5" cm"^(2)`

Answer

To find the increase in the surface area when a metallic cube is melted and recast into a cuboid, we will follow these steps:

### Step 1: Calculate the Volume of the Cube
The volume \( V \) of a cube is given by the formula:
\[ V = L^3 \]
where \( L \) is the edge length of the cube. In this case, the edge length \( L = 2.5 \) cm.

Calculating the volume:
\[ V = (2.5)^3 = 2.5 \times 2.5 \times 2.5 = 15.625 \text{ cm}^3 \]

### Step 2: Calculate the Height of the Cuboid
The volume of the cuboid is given by the formula:
\[ V = L \times B \times H \]
where \( L \) is the length, \( B \) is the breadth, and \( H \) is the height. The base dimensions of the cuboid are \( 1.25 \) cm and \( 0.25 \) cm.

Substituting the known values into the volume formula:
\[ 15.625 = 1.25 \times 0.25 \times H \]

Calculating the area of the base:
\[ 1.25 \times 0.25 = 0.3125 \text{ cm}^2 \]

Now substituting this back to find \( H \):
\[ 15.625 = 0.3125 \times H \]
\[ H = \frac{15.625}{0.3125} = 50 \text{ cm} \]

### Step 3: Calculate the Surface Area of the Cube
The surface area \( SA \) of a cube is given by:
\[ SA = 6L^2 \]
Substituting \( L = 2.5 \) cm:
\[ SA = 6 \times (2.5)^2 = 6 \times 6.25 = 37.5 \text{ cm}^2 \]

### Step 4: Calculate the Surface Area of the Cuboid
The surface area \( SA \) of a cuboid is given by:
\[ SA = 2(LB + BH + HL) \]
Substituting \( L = 1.25 \) cm, \( B = 0.25 \) cm, and \( H = 50 \) cm:
\[ SA = 2(1.25 \times 0.25 + 0.25 \times 50 + 50 \times 1.25) \]

Calculating each term:
1. \( LB = 1.25 \times 0.25 = 0.3125 \)
2. \( BH = 0.25 \times 50 = 12.5 \)
3. \( HL = 50 \times 1.25 = 62.5 \)

Now substituting back into the surface area formula:
\[ SA = 2(0.3125 + 12.5 + 62.5) = 2(75.3125) = 150.625 \text{ cm}^2 \]

### Step 5: Calculate the Increase in Surface Area
To find the increase in surface area, subtract the surface area of the cube from the surface area of the cuboid:
\[ \text{Increase} = SA_{\text{cuboid}} - SA_{\text{cube}} = 150.625 - 37.5 = 113.125 \text{ cm}^2 \]

### Final Answer
The increase in the surface area is:
\[ \boxed{113.125 \text{ cm}^2} \]