A rectangular block has length 10 cm, breadth 8 cm and height 2 cm. From this block , a cubical hole of side 2 cm is drilled out . Find the volume and the surface area of the remaining solid :

Options

A. `152 cm^(3) , 512 cm^(2) `
B. `125 cm^(3) , 215 cm^(2) `
C. `152 cm^(3) , 240 cm^(2) `
D. `125 cm^(3) , 512 cm^(2) `

Answer

To solve the problem, we need to find the volume and surface area of the remaining solid after drilling a cubical hole from a rectangular block. Let's break down the solution step by step.

### Step 1: Calculate the Volume of the Rectangular Block
The formula for the volume \( V \) of a rectangular block is given by:
\[ V = \text{Length} \times \text{Breadth} \times \text{Height} \]
Given:
- Length \( L = 10 \, \text{cm} \)
- Breadth \( B = 8 \, \text{cm} \)
- Height \( H = 2 \, \text{cm} \)

Calculating the volume:
\[ V = 10 \times 8 \times 2 = 160 \, \text{cm}^3 \]

### Step 2: Calculate the Volume of the Cubical Hole
The volume \( V_c \) of a cube is given by:
\[ V_c = \text{Side}^3 \]
Given:
- Side of the cube \( s = 2 \, \text{cm} \)

Calculating the volume of the cube:
\[ V_c = 2^3 = 8 \, \text{cm}^3 \]

### Step 3: Calculate the Remaining Volume
To find the remaining volume \( V_r \) after drilling the hole, we subtract the volume of the cube from the volume of the rectangular block:
\[ V_r = V - V_c = 160 - 8 = 152 \, \text{cm}^3 \]

### Step 4: Calculate the Surface Area of the Rectangular Block
The formula for the surface area \( SA \) of a rectangular block is given by:
\[ SA = 2(LB + BH + HL) \]
Calculating each term:
- \( LB = 10 \times 8 = 80 \)
- \( BH = 8 \times 2 = 16 \)
- \( HL = 2 \times 10 = 20 \)

Now substituting these values into the surface area formula:
\[ SA = 2(80 + 16 + 20) = 2(116) = 232 \, \text{cm}^2 \]

### Step 5: Adjust the Surface Area for the Drilled Hole
When a cube is drilled out, we lose the area of the faces that were part of the block but gain the area of the new faces created inside the hole. The cube has 6 faces, but since we are drilling it out, we will add back the area of 2 faces (the top and bottom of the hole).

The area of one face of the cube is:
\[ \text{Area of one face} = s^2 = 2^2 = 4 \, \text{cm}^2 \]
Thus, the area of two faces is:
\[ \text{Area of two faces} = 2 \times 4 = 8 \, \text{cm}^2 \]

Now, the new total surface area \( SA_r \) of the remaining solid is:
\[ SA_r = SA + \text{Area of two faces} = 232 + 8 = 240 \, \text{cm}^2 \]

### Final Results
- Remaining Volume: \( 152 \, \text{cm}^3 \)
- Remaining Surface Area: \( 240 \, \text{cm}^2 \)

### Summary
The volume of the remaining solid is \( 152 \, \text{cm}^3 \) and the surface area is \( 240 \, \text{cm}^2 \).