The circumference of the base of a cylinder is 176 cm and its height is 65 cm. find the volume of the cylinder and its lateral surface area.

Answer

To solve the problem, we need to find the volume and lateral surface area of a cylinder given its circumference and height. Here’s a step-by-step solution:

### Step 1: Find the radius of the cylinder
The formula for the circumference (C) of a cylinder is given by:
\[ C = 2\pi r \]
Where \( r \) is the radius.

We know the circumference is 176 cm. Therefore, we can set up the equation:
\[ 176 = 2\pi r \]

### Step 2: Solve for the radius (r)
We can rearrange the equation to solve for \( r \):
\[ r = \frac{176}{2\pi} \]

Using \( \pi \approx \frac{22}{7} \):
\[ r = \frac{176}{2 \times \frac{22}{7}} \]
\[ r = \frac{176 \times 7}{2 \times 22} \]
\[ r = \frac{1232}{44} \]
\[ r = 28 \, \text{cm} \]

### Step 3: Calculate the volume of the cylinder
The formula for the volume (V) of a cylinder is:
\[ V = \pi r^2 h \]
Where \( h \) is the height of the cylinder.

We know:
- \( r = 28 \, \text{cm} \)
- \( h = 65 \, \text{cm} \)

Substituting the values into the volume formula:
\[ V = \pi (28)^2 (65) \]
Using \( \pi \approx \frac{22}{7} \):
\[ V = \frac{22}{7} \times 28 \times 28 \times 65 \]

### Step 4: Simplify the volume calculation
First, calculate \( 28^2 \):
\[ 28^2 = 784 \]
Now substitute back into the volume formula:
\[ V = \frac{22}{7} \times 784 \times 65 \]

Next, we can simplify:
\[ V = \frac{22 \times 784 \times 65}{7} \]

Calculating \( \frac{784}{7} = 112 \):
\[ V = 22 \times 112 \times 65 \]

Now calculating \( 22 \times 112 = 2464 \):
\[ V = 2464 \times 65 \]

Finally, calculating \( 2464 \times 65 = 160160 \, \text{cm}^3 \).

### Step 5: Calculate the lateral surface area
The formula for the lateral surface area (LSA) of a cylinder is:
\[ LSA = 2\pi rh \]

Substituting the values:
\[ LSA = 2 \times \pi \times 28 \times 65 \]
Using \( \pi \approx \frac{22}{7} \):
\[ LSA = 2 \times \frac{22}{7} \times 28 \times 65 \]

### Step 6: Simplify the lateral surface area calculation
Calculating \( 2 \times 28 = 56 \):
\[ LSA = \frac{22}{7} \times 56 \times 65 \]

Calculating \( \frac{56}{7} = 8 \):
\[ LSA = 22 \times 8 \times 65 \]

Calculating \( 22 \times 8 = 176 \):
\[ LSA = 176 \times 65 \]

Finally, calculating \( 176 \times 65 = 11440 \, \text{cm}^2 \).

### Final Answers:
- Volume of the cylinder: **160160 cm³**
- Lateral Surface Area of the cylinder: **11440 cm²**