The total surface area of a cube is 32 2/3`m^(2)`. Find the volume of cube.

Answer

To find the volume of a cube given its total surface area, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Total Surface Area of a Cube**:
The total surface area (TSA) of a cube is given by the formula:
\[ \text{TSA} = 6a^2 \]
where \(a\) is the length of one side of the cube.

2. **Convert the Mixed Fraction to an Improper Fraction**:
The given total surface area is \(32 \frac{2}{3} \, m^2\). We need to convert this mixed number into an improper fraction:
\[ 32 \frac{2}{3} = 32 + \frac{2}{3} = \frac{32 \times 3 + 2}{3} = \frac{96 + 2}{3} = \frac{98}{3} \, m^2 \]

3. **Set Up the Equation**:
Now, we can set up the equation using the TSA formula:
\[ 6a^2 = \frac{98}{3} \]

4. **Solve for \(a^2\)**:
To isolate \(a^2\), divide both sides by 6:
\[ a^2 = \frac{98}{3} \div 6 = \frac{98}{3} \times \frac{1}{6} = \frac{98}{18} = \frac{49}{9} \]

5. **Find \(a\)**:
Now, take the square root of both sides to find \(a\):
\[ a = \sqrt{\frac{49}{9}} = \frac{\sqrt{49}}{\sqrt{9}} = \frac{7}{3} \, m \]

6. **Calculate the Volume of the Cube**:
The volume \(V\) of a cube is given by the formula:
\[ V = a^3 \]
Substitute \(a = \frac{7}{3}\):
\[ V = \left(\frac{7}{3}\right)^3 = \frac{7^3}{3^3} = \frac{343}{27} \, m^3 \]

7. **Convert to Mixed Fraction**:
To convert \(\frac{343}{27}\) into a mixed fraction:
- Divide 343 by 27, which gives 12 with a remainder of 19.
- Thus, \(\frac{343}{27} = 12 \frac{19}{27} \, m^3\).

### Final Answer:
The volume of the cube is:
\[ 12 \frac{19}{27} \, m^3 \]