Each side of a cube is measured to be 6.230 m. What is the total surface area and the volume of cube to appropriate significant figures?

Explanation

To solve the problem of finding the total surface area and volume of a cube with each side measuring 6.230 m, we will follow these steps:

### Step 1: Calculate the Total Surface Area of the Cube
The formula for the total surface area (SA) of a cube is given by:
\[ SA = 6a^2 \]
where \( a \) is the length of one side of the cube.

**Substituting the value of \( a \):**
\[ SA = 6 \times (6.230 \, \text{m})^2 \]

### Step 2: Calculate \( (6.230 \, \text{m})^2 \)
Calculating the square of the side length:
\[ (6.230)^2 = 38.870529 \, \text{m}^2 \]

### Step 3: Calculate the Total Surface Area
Now substituting back into the surface area formula:
\[ SA = 6 \times 38.870529 \, \text{m}^2 = 233.223174 \, \text{m}^2 \]

### Step 4: Round to Appropriate Significant Figures
The measurement 6.230 m has four significant figures. Therefore, we need to round the total surface area to four significant figures:
\[ SA \approx 233.2 \, \text{m}^2 \]

### Step 5: Calculate the Volume of the Cube
The formula for the volume (V) of a cube is given by:
\[ V = a^3 \]

**Substituting the value of \( a \):**
\[ V = (6.230 \, \text{m})^3 \]

### Step 6: Calculate \( (6.230 \, \text{m})^3 \)
Calculating the cube of the side length:
\[ (6.230)^3 = 243.101227 \, \text{m}^3 \]

### Step 7: Round to Appropriate Significant Figures
Again, since the measurement has four significant figures, we round the volume:
\[ V \approx 243.1 \, \text{m}^3 \]

### Final Results
- Total Surface Area: \( 233.2 \, \text{m}^2 \)
- Volume: \( 243.1 \, \text{m}^3 \)

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