Curved surfaceof a cylinderis `1000 cm^2` and the diameter of its base is 20 cm, then the volume of cylinder is:

Options

A. `5000 cm^(3)`
B. `4800 cm^(3)`
C. `395 cm^(3)`
D. `4000 cm^(3)`

Answer

To find the volume of the cylinder given its curved surface area and diameter, we can follow these steps:

### Step 1: Identify the given values
- Curved Surface Area (CSA) = 1000 cm²
- Diameter of the base = 20 cm

### Step 2: Calculate the radius of the base
The radius (R) can be calculated from the diameter using the formula:
\[ R = \frac{\text{Diameter}}{2} \]
Substituting the given diameter:
\[ R = \frac{20 \, \text{cm}}{2} = 10 \, \text{cm} \]

### Step 3: Use the formula for the curved surface area of a cylinder
The formula for the curved surface area (CSA) of a cylinder is:
\[ \text{CSA} = 2\pi R H \]
We can rearrange this formula to solve for the height (H):
\[ H = \frac{\text{CSA}}{2\pi R} \]
Substituting the values we have:
\[ H = \frac{1000 \, \text{cm}^2}{2\pi(10 \, \text{cm})} \]

### Step 4: Simplify to find the height (H)
Calculating the height:
\[ H = \frac{1000}{20\pi} = \frac{100}{2\pi} = \frac{50}{\pi} \, \text{cm} \]

### Step 5: Use the formula for the volume of a cylinder
The volume (V) of a cylinder is given by:
\[ V = \pi R^2 H \]
Substituting the values of R and H:
\[ V = \pi (10 \, \text{cm})^2 \left(\frac{50}{\pi} \, \text{cm}\right) \]

### Step 6: Calculate the volume
Calculating the volume:
\[ V = \pi \times 100 \times \frac{50}{\pi} \]
The \(\pi\) cancels out:
\[ V = 100 \times 50 = 5000 \, \text{cm}^3 \]

### Final Answer
The volume of the cylinder is \( 5000 \, \text{cm}^3 \).

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