A piece of rectangular cardboard sheet measuring 40 inch `xx25` inch is made into an open chocolate box by cutting out squares of side 'p' from each corner. Which of the following expressions is equivalent to the volume of the box ?

Options

A. `4p^(3)-120p^(2)+950p`
B. `4p^(3)+130p^(2)+1000p`
C. `4p^(3)-130p^(2)+1000p`
D. None of these

Answer

To find the volume of the open chocolate box made from the rectangular cardboard sheet, we follow these steps:

### Step 1: Understand the dimensions of the cardboard
The dimensions of the cardboard sheet are given as:
- Length = 40 inches
- Width = 25 inches

### Step 2: Identify the squares cut from each corner
We are cutting out squares of side 'p' from each corner of the cardboard. This will affect the dimensions of the box.

### Step 3: Calculate the new dimensions of the box
After cutting out squares of side 'p', the new dimensions of the box will be:
- **Length of the box**:
\[ \text{Length} = 40 - 2p \]
(since we cut 'p' from both ends of the length)

- **Width of the box**:
\[ \text{Width} = 25 - 2p \]
(since we cut 'p' from both ends of the width)

### Step 4: Determine the height of the box
The height of the box will be equal to the side of the square cut out, which is:
\[ \text{Height} = p \]

### Step 5: Write the formula for the volume of the box
The volume \( V \) of a box (cuboid) is given by the formula:
\[ V = \text{Length} \times \text{Width} \times \text{Height} \]
Substituting the dimensions we found:
\[ V = (40 - 2p)(25 - 2p)(p) \]

### Step 6: Expand the volume expression
Now we will expand this expression:
1. First, expand the first two terms:
\[ (40 - 2p)(25 - 2p) = 40 \times 25 - 40 \times 2p - 25 \times 2p + 2p \times 2p \]
\[ = 1000 - 80p - 50p + 4p^2 \]
\[ = 1000 - 130p + 4p^2 \]

2. Now multiply this result by \( p \):
\[ V = (1000 - 130p + 4p^2)(p) \]
\[ = 1000p - 130p^2 + 4p^3 \]

### Step 7: Write the final volume expression
Rearranging the terms, we get:
\[ V = 4p^3 - 130p^2 + 1000p \]

### Final Answer
The expression equivalent to the volume of the box is:
\[ V = 4p^3 - 130p^2 + 1000p \]