Each side of a cube is measured to be `7.203` m . What is (i) the total surface area and (ii) the volume of the cube to appropriate significant figures ?

Answer

To solve the problem, we need to calculate the total surface area and the volume of a cube with a side length of \(7.203\) m. We will also ensure that our answers are expressed to the appropriate significant figures.

### Step-by-Step Solution:

**Step 1: Calculate the Total Surface Area of the Cube**

The formula for the total surface area \(A\) of a cube with side length \(s\) is given by:

\[ A = 6s^2 \]

Substituting the given side length \(s = 7.203\) m into the formula:

\[ A = 6 \times (7.203)^2 \]

**Step 2: Calculate \(s^2\)**

First, we need to calculate \(s^2\):

\[ s^2 = (7.203)^2 = 51.883209 \]

**Step 3: Calculate the Total Surface Area**

Now we can calculate the total surface area:

\[ A = 6 \times 51.883209 = 311.299254 \, \text{m}^2 \]

**Step 4: Round to Appropriate Significant Figures**

The side length \(7.203\) m has four significant figures. Therefore, we should round the total surface area to four significant figures:

\[ A \approx 311.3 \, \text{m}^2 \]

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**Step 5: Calculate the Volume of the Cube**

The formula for the volume \(V\) of a cube with side length \(s\) is given by:

\[ V = s^3 \]

Substituting the given side length:

\[ V = (7.203)^3 \]

**Step 6: Calculate \(s^3\)**

Calculating \(s^3\):

\[ V = (7.203) \times (7.203) \times (7.203) = 373.260579 \, \text{m}^3 \]

**Step 7: Round to Appropriate Significant Figures**

Again, since the side length has four significant figures, we round the volume to four significant figures:

\[ V \approx 373.3 \, \text{m}^3 \]

### Final Answers:

(i) The total surface area of the cube is approximately \(311.3 \, \text{m}^2\).

(ii) The volume of the cube is approximately \(373.3 \, \text{m}^3\).

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