Two sheets of paper with measure22 cm by 28 cm are taken . Each sheetis then rolled intoa cylinder, one having the height 22 cm and the other of height 28 cm . The difference in their volumes (in ` cm^(3)` ) will be

Options

A. 216
B. 294
C. 316
D. 0

Answer

To find the difference in volumes of the two cylinders formed by rolling the sheets of paper, we will follow these steps:

### Step 1: Identify the dimensions of the sheets of paper
The dimensions of each sheet of paper are given as 22 cm by 28 cm.

### Step 2: Calculate the volume of the first cylinder
- **Height (h1)**: 22 cm (when the sheet is rolled along the length)
- **Circumference**: The circumference of the base will be equal to the other dimension, which is 28 cm.

Using the formula for circumference:
\[ C = 2\pi r \]
We can solve for the radius (r1):
\[ 2\pi r_1 = 28 \implies r_1 = \frac{28}{2\pi} = \frac{14}{\pi} \text{ cm} \]

Now, we can calculate the volume (V1) of the cylinder using the formula:
\[ V = \pi r^2 h \]
Substituting the values:
\[ V_1 = \pi \left(\frac{14}{\pi}\right)^2 \times 22 \]
\[ V_1 = \pi \left(\frac{196}{\pi^2}\right) \times 22 \]
\[ V_1 = \frac{196 \times 22}{\pi} = \frac{4312}{\pi} \text{ cm}^3 \]

### Step 3: Calculate the volume of the second cylinder
- **Height (h2)**: 28 cm (when the sheet is rolled along the width)
- **Circumference**: The circumference of the base will be equal to the other dimension, which is 22 cm.

Using the formula for circumference:
\[ C = 2\pi r \]
We can solve for the radius (r2):
\[ 2\pi r_2 = 22 \implies r_2 = \frac{22}{2\pi} = \frac{11}{\pi} \text{ cm} \]

Now, we can calculate the volume (V2) of the cylinder using the formula:
\[ V = \pi r^2 h \]
Substituting the values:
\[ V_2 = \pi \left(\frac{11}{\pi}\right)^2 \times 28 \]
\[ V_2 = \pi \left(\frac{121}{\pi^2}\right) \times 28 \]
\[ V_2 = \frac{121 \times 28}{\pi} = \frac{3388}{\pi} \text{ cm}^3 \]

### Step 4: Calculate the difference in volumes
Now we find the difference between the two volumes:
\[ \text{Difference} = V_1 - V_2 \]
Substituting the values:
\[ \text{Difference} = \frac{4312}{\pi} - \frac{3388}{\pi} = \frac{4312 - 3388}{\pi} = \frac{924}{\pi} \text{ cm}^3 \]

### Step 5: Final calculation
To get the numerical value, we can approximate \(\pi\) as 3.14:
\[ \text{Difference} \approx \frac{924}{3.14} \approx 294 \text{ cm}^3 \]

### Conclusion
The difference in volumes of the two cylinders is approximately **294 cm³**.

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