A cube of metal of 2.5 cm edge is melted and cast in to rectangular solid whose base is 1.25 cm by 0.25 cm .Assuming no loss in melting find the height of the solid . Also find the gain in the surface area.

Explanation

To solve the problem step by step, we will first find the volume of the cube, then use that volume to find the height of the rectangular solid, and finally calculate the surface areas to find the gain in surface area.

### Step 1: Calculate the volume of the cube
The volume \( V \) of a cube is given by the formula:
\[ V = \text{edge}^3 \]
Given that the edge of the cube is \( 2.5 \, \text{cm} \):
\[ V = (2.5)^3 = 2.5 \times 2.5 \times 2.5 \]
Calculating this:
\[ V = 2.5 \times 2.5 = 6.25 \]
\[ V = 6.25 \times 2.5 = 15.625 \, \text{cm}^3 \]

### Step 2: Calculate the area of the base of the rectangular solid
The area \( A \) of the base of the rectangular solid is given by:
\[ A = \text{length} \times \text{breadth} \]
Given that the base dimensions are \( 1.25 \, \text{cm} \) and \( 0.25 \, \text{cm} \):
\[ A = 1.25 \times 0.25 \]
Calculating this:
\[ A = 0.3125 \, \text{cm}^2 \]

### Step 3: Calculate the height of the rectangular solid
The height \( h \) of the rectangular solid can be found using the formula:
\[ h = \frac{V}{A} \]
Substituting the values we have:
\[ h = \frac{15.625}{0.3125} \]
Calculating this:
\[ h = 50 \, \text{cm} \]

### Step 4: Calculate the surface area of the cube
The surface area \( SA \) of a cube is given by:
\[ SA = 6 \times \text{edge}^2 \]
Substituting the edge length:
\[ SA = 6 \times (2.5)^2 \]
Calculating this:
\[ SA = 6 \times 6.25 = 37.5 \, \text{cm}^2 \]

### Step 5: Calculate the surface area of the rectangular solid
The surface area \( SA \) of a rectangular solid is given by:
\[ SA = 2(lb + bh + hl) \]
Where \( l \) is the length, \( b \) is the breadth, and \( h \) is the height. Substituting the values:
\[ SA = 2(1.25 \times 0.25 + 0.25 \times 50 + 50 \times 1.25) \]
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:
\[ SA = 2(0.3125 + 12.5 + 62.5) = 2(75.3125) = 150.625 \, \text{cm}^2 \]

### Step 6: Calculate the gain in surface area
The gain in surface area is given by:
\[ \text{Gain} = \text{Surface Area of Rectangular Solid} - \text{Surface Area of Cube} \]
Substituting the values:
\[ \text{Gain} = 150.625 - 37.5 = 113.125 \, \text{cm}^2 \]

### Final Results
- Height of the rectangular solid: \( 50 \, \text{cm} \)
- Gain in surface area: \( 113.125 \, \text{cm}^2 \)