The total surface area of solid cylinder is `231 cm^(2)` and its curved surface area is `(2)/(3)` of the total surface area. Find the volume of the cylinder.

Answer

To solve the problem step by step, we will follow the given information and formulas related to the total surface area and volume of a cylinder.

### Step 1: Understand the formulas
The total surface area (TSA) of a cylinder is given by the formula:
\[ \text{TSA} = 2\pi r(h + r) \]
The curved surface area (CSA) of a cylinder is given by:
\[ \text{CSA} = 2\pi rh \]

### Step 2: Set up the equations
We know from the problem statement:
- The total surface area (TSA) is \( 231 \, \text{cm}^2 \).
- The curved surface area (CSA) is \( \frac{2}{3} \) of the total surface area.

From this, we can write:
1. \( 2\pi r(h + r) = 231 \) (Equation 1)
2. \( 2\pi rh = \frac{2}{3} \times 231 \) (Equation 2)

### Step 3: Simplify Equation 2
Calculating the CSA:
\[ 2\pi rh = \frac{2}{3} \times 231 = 154 \]
So, we have:
\[ 2\pi rh = 154 \]

### Step 4: Solve for h in terms of r
From Equation 2:
\[ h = \frac{154}{2\pi r} \]
Substituting \( \pi \) as \( \frac{22}{7} \):
\[ h = \frac{154}{2 \times \frac{22}{7} \times r} = \frac{154 \times 7}{44r} = \frac{1078}{44r} = \frac{539}{22r} \]

### Step 5: Substitute h in Equation 1
Now substitute \( h \) back into Equation 1:
\[ 2\pi r \left( \frac{539}{22r} + r \right) = 231 \]
This simplifies to:
\[ 2\pi r \left( \frac{539 + 22r^2}{22r} \right) = 231 \]
Multiplying through by \( 22r \):
\[ 2\pi(539 + 22r^2) = 231 \times 22r \]

### Step 6: Solve for r
Substituting \( \pi \) as \( \frac{22}{7} \):
\[ 2 \times \frac{22}{7}(539 + 22r^2) = 231 \times 22r \]
This simplifies to:
\[ \frac{44}{7}(539 + 22r^2) = 231 \times 22r \]
Cross-multiplying gives:
\[ 44(539 + 22r^2) = 231 \times 154r \]
Now, solving for \( r \):
\[ 44 \times 539 + 968r^2 = 35454r \]
\[ 968r^2 - 35454r + 23716 = 0 \]

### Step 7: Use the quadratic formula
Using the quadratic formula \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Where \( a = 968, b = -35454, c = 23716 \).

### Step 8: Calculate the volume
Once we find \( r \), we can find \( h \) using \( h = 2r \) and then calculate the volume using:
\[ V = \pi r^2 h \]

### Final Calculation
Substituting the values of \( r \) and \( h \) into the volume formula will give us the final volume of the cylinder.