If volume of cylindrical vessel is `3696 cm^(3)`and ratio of height and radius of cylindrical vessel is 3:7, then find total surface area of cylindrical vessel.

Options

A. A)`1590cm ^(2)`
B. B)`1270cm ^(2)`
C. C)`1450 cm ^(2)`
D. D)`1760 cm ^(2)`

Answer

To find the total surface area of the cylindrical vessel, we will follow these steps:

### Step 1: Understand the given information
We know the volume of the cylindrical vessel is \( 3696 \, \text{cm}^3 \) and the ratio of height (H) to radius (R) is \( 3:7 \).

### Step 2: Express height and radius in terms of a variable
Let the height \( H = 3k \) and the radius \( R = 7k \), where \( k \) is a common multiplier.

### Step 3: Write the formula for the volume of a cylinder
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi R^2 H \]
Substituting the expressions for \( R \) and \( H \):
\[ 3696 = \pi (7k)^2 (3k) \]

### Step 4: Substitute the value of \( \pi \)
Using \( \pi \approx \frac{22}{7} \):
\[ 3696 = \frac{22}{7} (7k)^2 (3k) \]
This simplifies to:
\[ 3696 = \frac{22}{7} \cdot 49k^2 \cdot 3k \]
\[ 3696 = \frac{22 \cdot 147k^3}{7} \]

### Step 5: Simplify the equation
Multiply both sides by 7 to eliminate the fraction:
\[ 3696 \times 7 = 22 \cdot 147k^3 \]
Calculating \( 3696 \times 7 \):
\[ 25872 = 22 \cdot 147k^3 \]

### Step 6: Solve for \( k^3 \)
Now divide both sides by \( 22 \cdot 147 \):
\[ k^3 = \frac{25872}{22 \cdot 147} \]
Calculating \( 22 \cdot 147 = 3234 \):
\[ k^3 = \frac{25872}{3234} = 8 \]
Taking the cube root:
\[ k = 2 \]

### Step 7: Find the height and radius
Now substitute \( k \) back to find \( H \) and \( R \):
\[ H = 3k = 3 \times 2 = 6 \, \text{cm} \]
\[ R = 7k = 7 \times 2 = 14 \, \text{cm} \]

### Step 8: Calculate the total surface area
The total surface area \( A \) of a cylinder is given by:
\[ A = 2\pi R (H + R) \]
Substituting the values of \( R \) and \( H \):
\[ A = 2 \cdot \frac{22}{7} \cdot 14 \cdot (6 + 14) \]
Calculating \( (6 + 14) = 20 \):
\[ A = 2 \cdot \frac{22}{7} \cdot 14 \cdot 20 \]
Calculating step by step:
\[ A = 2 \cdot \frac{22 \cdot 14 \cdot 20}{7} \]
\[ A = \frac{2 \cdot 22 \cdot 14 \cdot 20}{7} = \frac{12320}{7} = 1760 \, \text{cm}^2 \]

### Final Answer
The total surface area of the cylindrical vessel is \( 1760 \, \text{cm}^2 \).

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