Find the volume of the largest cylinder formed when a rectangular piece of paper 44 cm by 33 cm is rolled along its shorter side.

Answer

To find the volume of the largest cylinder formed when a rectangular piece of paper measuring 44 cm by 33 cm is rolled along its shorter side, we can follow these steps:

### Step 1: Identify the dimensions of the rectangular paper
The dimensions of the rectangular piece of paper are:
- Length (L) = 44 cm
- Width (W) = 33 cm

### Step 2: Determine the circumference of the base of the cylinder
When the paper is rolled along its shorter side (which is 33 cm), the circumference (C) of the base of the cylinder is equal to the width of the paper:
\[ C = 33 \text{ cm} \]

### Step 3: Relate the circumference to the radius
The circumference of a circle is given by the formula:
\[ C = 2\pi r \]
Where \( r \) is the radius of the cylinder. We can set this equal to the circumference we found:
\[ 2\pi r = 33 \]
Now, solve for \( r \):
\[ r = \frac{33}{2\pi} \]

### Step 4: Determine the height of the cylinder
The height (h) of the cylinder is equal to the length of the paper, which is:
\[ h = 44 \text{ cm} \]

### Step 5: Calculate the volume of the cylinder
The volume (V) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
Substituting the values of \( r \) and \( h \):
\[ V = \pi \left(\frac{33}{2\pi}\right)^2 \times 44 \]

### Step 6: Simplify the expression
Calculating \( r^2 \):
\[ r^2 = \left(\frac{33}{2\pi}\right)^2 = \frac{1089}{4\pi^2} \]
Now substituting back into the volume formula:
\[ V = \pi \times \frac{1089}{4\pi^2} \times 44 \]
\[ V = \frac{1089 \times 44}{4\pi} \]

### Step 7: Calculate the final volume
Now, calculate the volume:
\[ V = \frac{47856}{4\pi} \]
\[ V = \frac{11964}{\pi} \]

### Step 8: Approximate the volume using \( \pi \approx 3.14 \)
Using \( \pi \approx 3.14 \):
\[ V \approx \frac{11964}{3.14} \approx 3815.29 \text{ cm}^3 \]

Thus, the volume of the largest cylinder formed is approximately \( 3815.29 \text{ cm}^3 \).

---