The ratio between the radius of the base and the height of a cylindrical pillar is 3:4 . If its volume is ` 4851 m^(3) `, the curved surface area of the pillar is :

Options

A. `924 m^(2) `
B. `1617 m^(2) `
C. `425 m^(2)`
D. none of these

Answer

To find the curved surface area of a cylindrical pillar given the ratio of the radius to height and the volume, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Ratio**:
The ratio of the radius (r) to the height (h) of the cylindrical pillar is given as 3:4.
Let the radius be \( r = 3x \) and the height be \( h = 4x \).

2. **Volume of the Cylinder**:
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
We know the volume is \( 4851 \, m^3 \). Substituting the values of \( r \) and \( h \):
\[ 4851 = \pi (3x)^2 (4x) \]

3. **Substituting Values**:
Substitute \( \pi \) with \( \frac{22}{7} \):
\[ 4851 = \frac{22}{7} (3x)^2 (4x) \]
Simplifying further:
\[ 4851 = \frac{22}{7} \cdot 9x^2 \cdot 4x \]
\[ 4851 = \frac{22 \cdot 36x^3}{7} \]

4. **Cross Multiplying**:
To eliminate the fraction, cross-multiply:
\[ 4851 \cdot 7 = 22 \cdot 36x^3 \]
\[ 33957 = 792x^3 \]

5. **Solving for \( x^3 \)**:
Divide both sides by 792:
\[ x^3 = \frac{33957}{792} \]
Calculating the right-hand side:
\[ x^3 = 42.875 \]

6. **Finding \( x \)**:
Take the cube root of both sides:
\[ x = \sqrt[3]{42.875} \approx 3.5 \, m \]

7. **Finding Radius and Height**:
Now, substitute \( x \) back to find the radius and height:
\[ r = 3x = 3 \cdot 3.5 = 10.5 \, m \]
\[ h = 4x = 4 \cdot 3.5 = 14 \, m \]

8. **Curved Surface Area of the Cylinder**:
The formula for the curved surface area (CSA) of a cylinder is:
\[ CSA = 2\pi rh \]
Substituting the values of \( r \) and \( h \):
\[ CSA = 2 \cdot \frac{22}{7} \cdot 10.5 \cdot 14 \]

9. **Calculating CSA**:
Simplifying the expression:
\[ CSA = 2 \cdot \frac{22}{7} \cdot 10.5 \cdot 14 = 2 \cdot \frac{22 \cdot 10.5 \cdot 14}{7} \]
\[ = 2 \cdot \frac{3084}{7} = 2 \cdot 440.5714 \approx 924 \, m^2 \]

### Final Answer:
The curved surface area of the cylindrical pillar is \( 924 \, m^2 \).

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