A square hole of cross - sectional area `4" cm"^(2)` is drilled across a cube with its length parallel to a side of the cube. If an edge of the cube measures 5 cm, what is the total surface area of the body so formed ?

Options

A. `140" cm"^(2)`
B. `142" cm"^(2)`
C. `162" cm"^(2)`
D. `182" cm"^(2)`

Answer

To find the total surface area of the body formed after drilling a square hole through a cube, we can follow these steps:

### Step 1: Calculate the Surface Area of the Cube
The formula for the surface area of a cube is given by:
\[ \text{Surface Area} = 6a^2 \]
where \(a\) is the length of an edge of the cube. Given that the edge of the cube measures \(5 \, \text{cm}\):
\[ \text{Surface Area} = 6 \times (5 \, \text{cm})^2 = 6 \times 25 \, \text{cm}^2 = 150 \, \text{cm}^2 \]

### Step 2: Calculate the Area of the Square Hole
The area of the square hole is given as \(4 \, \text{cm}^2\). Since the hole is drilled through the cube, it will remove the area from two faces of the cube (the entrance and exit of the hole). Therefore, we subtract this area twice:
\[ \text{Area removed} = 2 \times 4 \, \text{cm}^2 = 8 \, \text{cm}^2 \]

### Step 3: Calculate the Surface Area After Removing the Hole
Now, we subtract the area removed from the total surface area of the cube:
\[ \text{New Surface Area} = \text{Original Surface Area} - \text{Area removed} = 150 \, \text{cm}^2 - 8 \, \text{cm}^2 = 142 \, \text{cm}^2 \]

### Step 4: Calculate the Surface Area of the Walls of the Hole
The hole has four walls, and we need to calculate the area of these walls. The height of the hole is equal to the length of the cube, which is \(5 \, \text{cm}\). The perimeter of the square hole (which is the total length of the walls) is:
\[ \text{Perimeter} = 4 \times \text{side length} \]
To find the side length of the hole, we use the area:
\[ \text{Area} = \text{side}^2 \implies \text{side} = \sqrt{4 \, \text{cm}^2} = 2 \, \text{cm} \]
Thus, the perimeter is:
\[ \text{Perimeter} = 4 \times 2 \, \text{cm} = 8 \, \text{cm} \]
The area of the walls of the hole is:
\[ \text{Area of walls} = \text{Perimeter} \times \text{height} = 8 \, \text{cm} \times 5 \, \text{cm} = 40 \, \text{cm}^2 \]

### Step 5: Calculate the Total Surface Area of the Body Formed
Finally, we add the area of the walls of the hole to the new surface area:
\[ \text{Total Surface Area} = \text{New Surface Area} + \text{Area of walls} = 142 \, \text{cm}^2 + 40 \, \text{cm}^2 = 182 \, \text{cm}^2 \]

### Final Answer
The total surface area of the body formed is:
\[ \boxed{182 \, \text{cm}^2} \]