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

Answer

To find the thickness of the hollow square-shaped tube made of iron, 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. We need to find the thickness of the tube, which we will denote as \( x \).

### Step 2: Determine the dimensions of the external square
The side length of the external square can be expressed as:
\[ \text{Side of external square} = \text{Side of internal square} + 2 \times \text{thickness} = 5 + 2x \]

### Step 3: Calculate the volume of the external square
The volume of the external square tube can be calculated using the formula for the volume of a rectangular prism:
\[ \text{Volume of external square} = (\text{Side of external square})^2 \times \text{Length} = (5 + 2x)^2 \times 8 \]

### Step 4: Calculate the volume of the internal square
The volume of the internal square tube is:
\[ \text{Volume of internal square} = (\text{Side of internal square})^2 \times \text{Length} = 5^2 \times 8 = 25 \times 8 = 200 \, \text{cm}^3 \]

### Step 5: Set up the equation for the volume of iron
The volume of iron in the tube is given as 192 cm³. Therefore, we can set up the equation:
\[ \text{Volume of iron} = \text{Volume of external square} - \text{Volume of internal square} \]
This gives us:
\[ 192 = (5 + 2x)^2 \times 8 - 200 \]

### Step 6: Simplify the equation
Rearranging the equation:
\[ (5 + 2x)^2 \times 8 = 192 + 200 \]
\[ (5 + 2x)^2 \times 8 = 392 \]
Dividing both sides by 8:
\[ (5 + 2x)^2 = 49 \]

### Step 7: Solve for \( x \)
Taking the square root of both sides:
\[ 5 + 2x = 7 \quad \text{or} \quad 5 + 2x = -7 \]
From \( 5 + 2x = 7 \):
\[ 2x = 7 - 5 = 2 \quad \Rightarrow \quad x = 1 \]
From \( 5 + 2x = -7 \):
\[ 2x = -7 - 5 = -12 \quad \Rightarrow \quad x = -6 \quad \text{(not possible)} \]

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