The volume of a cylinder is 448 `pi cm^3` and height 7 cm. Then, Its lateral surface area and total surface area Is

Options

A. `352cm^2, 754.286cm^2`
B. `252 cm^2, 755.286 cm^2`
C. `259cm^2,457.206cm^2`
D. None of the above

Answer

To solve the problem, we need to find the lateral surface area and total surface area of a cylinder given its volume and height. Let's go through the steps one by one.

### Step 1: Find the Radius of the Cylinder
The formula for the volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
We know the volume \( V = 448\pi \, \text{cm}^3 \) and the height \( h = 7 \, \text{cm} \).

Setting the volume formula equal to the given volume:
\[ \pi r^2 h = 448\pi \]
We can cancel \( \pi \) from both sides:
\[ r^2 h = 448 \]
Substituting the height \( h = 7 \):
\[ r^2 \cdot 7 = 448 \]
Now, divide both sides by 7:
\[ r^2 = \frac{448}{7} = 64 \]
Taking the square root of both sides gives:
\[ r = \sqrt{64} = 8 \, \text{cm} \]

### Step 2: Calculate the Lateral Surface Area
The formula for the lateral surface area \( A_L \) of a cylinder is:
\[ A_L = 2\pi rh \]
Substituting the values of \( r \) and \( h \):
\[ A_L = 2\pi (8)(7) \]
Calculating this:
\[ A_L = 2 \cdot \frac{22}{7} \cdot 8 \cdot 7 \]
The \( 7 \) cancels out:
\[ A_L = 2 \cdot 22 \cdot 8 = 352 \, \text{cm}^2 \]

### Step 3: Calculate the Total Surface Area
The formula for the total surface area \( A_T \) of a cylinder is:
\[ A_T = 2\pi r (r + h) \]
Substituting the values of \( r \) and \( h \):
\[ A_T = 2\pi (8)(8 + 7) \]
Calculating this:
\[ A_T = 2\pi (8)(15) \]
Now substituting \( \pi \) with \( \frac{22}{7} \):
\[ A_T = 2 \cdot \frac{22}{7} \cdot 8 \cdot 15 \]
Calculating further:
\[ A_T = \frac{44}{7} \cdot 120 = \frac{5280}{7} \approx 754.286 \, \text{cm}^2 \]

### Final Answers
- Lateral Surface Area: \( 352 \, \text{cm}^2 \)
- Total Surface Area: \( \approx 754.286 \, \text{cm}^2 \)