The ratio of radius of the base and the heightof solid right circular cylinderis 2 : 3. If its volume is 12936 `"cm"^(3)`, then its total surface area is : (Take `pi=(22)/(7)` )

Options

A. 1. `2864" cm"^(2)`
B. 2. `3080" cm"^(2)`
C. 3. `3120" cm"^(2)`
D. 4. `3040" cm"^(2)`

Answer

To solve the problem step by step, we will follow the given information about the cylinder and apply the relevant formulas.

### Step 1: Understand the given ratio
The ratio of the radius (r) of the base to the height (h) of the cylinder is given as 2:3. We can express this as:
- Let the radius \( r = 2x \)
- Let the height \( h = 3x \)

### Step 2: Write the formula for the volume of the cylinder
The volume \( V \) of a right circular cylinder is given by the formula:
\[ V = \pi r^2 h \]
Substituting the expressions for \( r \) and \( h \):
\[ V = \pi (2x)^2 (3x) \]

### Step 3: Simplify the volume expression
Calculating \( (2x)^2 \):
\[ (2x)^2 = 4x^2 \]
Now substituting this back into the volume formula:
\[ V = \pi \cdot 4x^2 \cdot 3x = 12\pi x^3 \]

### Step 4: Substitute the given volume
We know from the problem that the volume \( V = 12936 \, \text{cm}^3 \). Thus, we can set up the equation:
\[ 12\pi x^3 = 12936 \]
Substituting \( \pi = \frac{22}{7} \):
\[ 12 \cdot \frac{22}{7} x^3 = 12936 \]

### Step 5: Solve for \( x^3 \)
First, simplify the left side:
\[ \frac{264}{7} x^3 = 12936 \]
Now, multiply both sides by 7 to eliminate the fraction:
\[ 264 x^3 = 12936 \cdot 7 \]
Calculating \( 12936 \cdot 7 \):
\[ 12936 \cdot 7 = 90552 \]
Now divide both sides by 264:
\[ x^3 = \frac{90552}{264} \]
Calculating \( \frac{90552}{264} \):
\[ x^3 = 343 \]
Taking the cube root:
\[ x = 7 \]

### Step 6: Find the radius and height
Now we can find the radius and height using \( x = 7 \):
- Radius \( r = 2x = 2 \cdot 7 = 14 \, \text{cm} \)
- Height \( h = 3x = 3 \cdot 7 = 21 \, \text{cm} \)

### Step 7: Calculate the total surface area
The total surface area \( A \) of a cylinder is given by the formula:
\[ A = 2\pi r(h + r) \]
Substituting the values of \( r \) and \( h \):
\[ A = 2 \cdot \frac{22}{7} \cdot 14 \cdot (21 + 14) \]
Calculating \( (21 + 14) = 35 \):
\[ A = 2 \cdot \frac{22}{7} \cdot 14 \cdot 35 \]
Calculating \( 2 \cdot 14 = 28 \):
\[ A = \frac{22}{7} \cdot 28 \cdot 35 \]
Calculating \( 28 \cdot 35 = 980 \):
\[ A = \frac{22 \cdot 980}{7} \]
Calculating \( \frac{980}{7} = 140 \):
\[ A = 22 \cdot 140 = 3080 \, \text{cm}^2 \]

### Final Answer
The total surface area of the cylinder is \( 3080 \, \text{cm}^2 \).

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