The volume of a cylinder is `48.125 cm^3`, which is formed by rolling a rectangular paper sheet along the length of the paper. If a cuboidal box without any lid i.e., open at the top` is made from the same sheet of paper by cutting out the square of side 0.5 cm from each of the four corners of the paper sheet, then what is the volume of this box?

Options

A. `20 cm^3`
B. `38 cm^3`
C. `19 cm^3`
D. none of these

Answer

To find the volume of the cuboidal box formed from the rectangular paper sheet, we will follow these steps:

### Step 1: Understand the volume of the cylinder
The volume of the cylinder is given as \( V = 48.125 \, \text{cm}^3 \). The formula for the volume of a cylinder is:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.

### Step 2: Rearrange the formula
We can rearrange the formula to find \( r^2 h \):
\[ r^2 h = \frac{V}{\pi} \]
Substituting the value of \( V \) and using \( \pi \approx \frac{22}{7} \):
\[ r^2 h = \frac{48.125 \times 7}{22} \]

### Step 3: Calculate \( r^2 h \)
Calculating the right side:
\[ r^2 h = \frac{337.875}{22} \approx 15.3125 \]

### Step 4: Assume a height for the cylinder
Assume \( h = 5 \, \text{cm} \) (as a reasonable height for the cylinder). Then we can find \( r^2 \):
\[ r^2 = \frac{15.3125}{5} = 3.0625 \]

### Step 5: Calculate the radius
Taking the square root to find \( r \):
\[ r = \sqrt{3.0625} \approx 1.75 \, \text{cm} \]

### Step 6: Find the dimensions of the rectangular sheet
The length of the rectangular sheet used to form the cylinder is given by the circumference of the base of the cylinder:
\[ \text{Length} = 2\pi r = 2 \times \frac{22}{7} \times 1.75 \approx 11 \, \text{cm} \]
The breadth of the sheet is the height of the cylinder, which we assumed to be \( 5 \, \text{cm} \).

### Step 7: Adjust dimensions for the cuboidal box
When we cut out squares of side \( 0.5 \, \text{cm} \) from each corner, the new dimensions of the box will be:
- New Length = \( 11 - 2 \times 0.5 = 10 \, \text{cm} \)
- New Breadth = \( 5 - 2 \times 0.5 = 4 \, \text{cm} \)
- Height = \( 0.5 \, \text{cm} \)

### Step 8: Calculate the volume of the box
The volume \( V \) of the cuboidal box is given by:
\[ V = \text{Length} \times \text{Breadth} \times \text{Height} = 10 \times 4 \times 0.5 \]
Calculating this gives:
\[ V = 20 \, \text{cm}^3 \]

Thus, the volume of the cuboidal box is \( 20 \, \text{cm}^3 \).