The volume of a solid cylinder is `96228 cm^(3)` and the ratio of its radius to its height is 9:14. Find the total surface area of the cylinder.

Answer

To solve the problem step by step, we need to find the total surface area of a cylinder given its volume and the ratio of its radius to height.

### Step 1: Understand the given information
We are given:
- Volume of the cylinder, \( V = 96228 \, \text{cm}^3 \)
- Ratio of radius to height, \( r:h = 9:14 \)

### Step 2: Express radius and height in terms of a variable
Let the radius \( r = 9x \) and the height \( h = 14x \), where \( x \) is a common factor.

### 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 \( h \):
\[ 96228 = \pi (9x)^2 (14x) \]

### Step 4: Substitute the value of \( \pi \)
Using \( \pi \approx \frac{22}{7} \):
\[ 96228 = \frac{22}{7} (9x)^2 (14x) \]

### Step 5: Simplify the equation
Calculating \( (9x)^2 \):
\[ (9x)^2 = 81x^2 \]
Thus, we have:
\[ 96228 = \frac{22}{7} \cdot 81x^2 \cdot 14x \]
\[ = \frac{22 \cdot 81 \cdot 14}{7} x^3 \]

### Step 6: Calculate the constant factor
Calculating \( 22 \cdot 81 \cdot 14 \):
\[ 22 \cdot 81 = 1782 \]
\[ 1782 \cdot 14 = 24948 \]
Thus, we have:
\[ 96228 = \frac{24948}{7} x^3 \]

### Step 7: Solve for \( x^3 \)
Multiply both sides by 7:
\[ 96228 \cdot 7 = 24948 x^3 \]
Calculating \( 96228 \cdot 7 \):
\[ 674596 = 24948 x^3 \]
Now, divide both sides by 24948:
\[ x^3 = \frac{674596}{24948} = 27 \]

### Step 8: Find the value of \( x \)
Taking the cube root:
\[ x = 3 \]

### Step 9: Calculate the radius and height
Now substituting back to find \( r \) and \( h \):
\[ r = 9x = 9 \cdot 3 = 27 \, \text{cm} \]
\[ h = 14x = 14 \cdot 3 = 42 \, \text{cm} \]

### Step 10: Calculate the total surface area
The total surface area \( A \) of a cylinder is given by:
\[ A = 2\pi r (r + h) \]
Substituting the values of \( r \) and \( h \):
\[ A = 2 \cdot \frac{22}{7} \cdot 27 \cdot (27 + 42) \]
Calculating \( 27 + 42 = 69 \):
\[ A = 2 \cdot \frac{22}{7} \cdot 27 \cdot 69 \]

### Step 11: Calculate the total surface area
Calculating:
\[ A = \frac{2 \cdot 22 \cdot 27 \cdot 69}{7} \]
Calculating \( 2 \cdot 22 \cdot 27 \cdot 69 \):
\[ = 81,972 \]
Thus:
\[ A = \frac{81972}{7} \approx 11710.28 \, \text{cm}^2 \]

### Final Answer:
The total surface area of the cylinder is approximately \( 11710.28 \, \text{cm}^2 \).

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