A hollow square-shaped tube open at both ends is made of iron. The internal square is of 5cm side and the length of the tube is 8cm. There are `192cm^(3)` of iron in this tube. Find its thickness.

Answer

To find the thickness of the hollow square-shaped tube, we will follow these steps:

### Step 1: Understand the dimensions
The internal square has a side length of 5 cm, and the length of the tube is 8 cm. Let the thickness of the tube be \( x \) cm.

### Step 2: Determine the external dimensions
The external side of the square will be the internal side plus twice the thickness (once for each side). Therefore, the external side length can be expressed as:
\[ \text{External side} = 5 + 2x \text{ cm} \]

### Step 3: Calculate the volumes
The volume of the iron in the tube is given as \( 192 \, \text{cm}^3 \). The volume of the tube can be calculated by subtracting the internal volume from the external volume.

- **External Volume**:
\[ \text{External Volume} = (\text{External side})^2 \times \text{Length} = (5 + 2x)^2 \times 8 \]

- **Internal Volume**:
\[ \text{Internal Volume} = (\text{Internal side})^2 \times \text{Length} = 5^2 \times 8 = 25 \times 8 = 200 \, \text{cm}^3 \]

### Step 4: Set up the equation
The volume of iron can be expressed as:
\[ \text{External Volume} - \text{Internal Volume} = 192 \]
Substituting the volumes we calculated:
\[ (5 + 2x)^2 \times 8 - 200 = 192 \]

### Step 5: Simplify the equation
Rearranging the equation:
\[ (5 + 2x)^2 \times 8 = 392 \]
Dividing both sides by 8:
\[ (5 + 2x)^2 = 49 \]

### Step 6: Solve for \( x \)
Taking the square root of both sides:
\[ 5 + 2x = 7 \quad \text{or} \quad 5 + 2x = -7 \quad (\text{not valid since thickness cannot be negative}) \]
Solving for \( x \):
\[ 2x = 7 - 5 \\ 2x = 2 \\ x = 1 \, \text{cm} \]

### Conclusion
The thickness of the tube is:
\[ \text{Thickness} = 1 \, \text{cm} \]