A cylinder has a diameter of 20 cm. The area of the curved surface is 100 `cm^( 2)`. Find the volume of the cylinder (in. `cm^( 3)`).

Answer

To find the volume of the cylinder, we can follow these steps:

### Step 1: Find the radius of the cylinder
The diameter of the cylinder is given as 20 cm. The radius (r) is half of the diameter.

\[ r = \frac{\text{diameter}}{2} = \frac{20 \text{ cm}}{2} = 10 \text{ cm} \]

**Hint:** Remember that the radius is always half of the diameter.

### Step 2: Use the formula for the curved surface area of the cylinder
The formula for the curved surface area (CSA) of a cylinder is given by:

\[ \text{CSA} = 2 \pi r h \]

We know the CSA is 100 cm², and we have found the radius (r = 10 cm). We can substitute these values into the formula to find the height (h).

\[ 100 = 2 \pi (10) h \]

### Step 3: Solve for height (h)
Rearranging the equation to solve for h:

\[ 100 = 20 \pi h \]

Dividing both sides by \(20 \pi\):

\[ h = \frac{100}{20 \pi} = \frac{5}{\pi} \text{ cm} \]

**Hint:** When rearranging equations, make sure to isolate the variable you are solving for.

### Step 4: Use the formula for the volume of the cylinder
The formula for the volume (V) of a cylinder is given by:

\[ V = \pi r^2 h \]

Substituting the values of r and h into the volume formula:

\[ V = \pi (10)^2 \left(\frac{5}{\pi}\right) \]

### Step 5: Calculate the volume
Calculating \(10^2\):

\[ V = \pi (100) \left(\frac{5}{\pi}\right) \]

The \(\pi\) cancels out:

\[ V = 100 \times 5 = 500 \text{ cm}^3 \]

### Final Answer
The volume of the cylinder is \(500 \text{ cm}^3\).

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