If the radius of the base of a right circular cylinder is 4 m and the area of the curved surface of the cylinder is 19.5 `m^2` then its volume in `m^3`

Options

A. 65
B. 42
C. 36
D. 39

Answer

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

### Step 1: Identify the given values
- Radius \( r = 4 \) m
- Curved Surface Area (CSA) \( = 19.5 \) m²

### Step 2: Use the formula for Curved Surface Area of a Cylinder
The formula for the Curved Surface Area (CSA) of a cylinder is given by:
\[ \text{CSA} = 2 \pi r h \]
where \( r \) is the radius and \( h \) is the height of the cylinder.

### Step 3: Substitute the known values into the CSA formula
Substituting the values we have:
\[ 19.5 = 2 \pi (4) h \]
This simplifies to:
\[ 19.5 = 8 \pi h \]

### Step 4: Solve for height \( h \)
To find \( h \), we can rearrange the equation:
\[ h = \frac{19.5}{8 \pi} \]

### Step 5: Calculate \( h \)
Using the approximate value of \( \pi \approx 3.14 \):
\[ h = \frac{19.5}{8 \times 3.14} \approx \frac{19.5}{25.12} \approx 0.774 \text{ m} \]

### Step 6: Use the formula for the volume of a cylinder
The formula for the volume \( V \) of a cylinder is:
\[ V = \pi r^2 h \]

### Step 7: Substitute the values of \( r \) and \( h \) into the volume formula
Substituting the values we have:
\[ V = \pi (4^2) \left(\frac{19.5}{8 \pi}\right) \]
This simplifies to:
\[ V = \pi (16) \left(\frac{19.5}{8 \pi}\right) \]

### Step 8: Cancel \( \pi \) and simplify
\[ V = 16 \left(\frac{19.5}{8}\right) \]
\[ V = \frac{16 \times 19.5}{8} \]

### Step 9: Calculate the volume
\[ V = \frac{312}{8} = 39 \text{ m}^3 \]

### Final Answer
The volume of the cylinder is \( 39 \text{ m}^3 \).

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