Each side of a cube is measured to be 7.203 m. What are the total surface area and the volume of the cube to appropriate significant figures ?

The correct option is 373.7 m³.

Step-by-Step Explanation:

1. Identify the given value and significant figures:
The side of the cube is given as:
$$a=7.203\text{m}$$
The number 7.203 has four significant figures (7, 2, 0, and 3).

2. Calculate the Volume of the Cube:
The formula for the volume ($V$) of a cube is:
$$V=a^3$$
Substituting the given value of the side length:
$$V={\left(7.203\text{m}\right)}^3$$
$$V=7.203\times 7.203\times 7.203{\text{m}}^3=373.71475{\text{m}}^3$$

3. Apply rules for Significant Figures in multiplication:
According to the rules of arithmetic operations with significant figures, the final result of a multiplication or power operation should contain the same number of significant figures as the original measurement with the least number of significant figures.
Since our measurement 7.203 m has 4 significant figures, the calculated volume must also be rounded to 4 significant figures:
$$V\approx 373.7{\text{m}}^3$$

4. Calculate the Total Surface Area (for completeness):
The formula for the total surface area ($A$) of a cube is:
$$A=6a^2$$
Substituting the side length:
$$A=6\times {\left(7.203\text{m}\right)}^2$$
$$A=6\times 51.883209{\text{m}}^2=311.299254{\text{m}}^2$$
Since the number 6 is an exact constant and has no uncertainty, it does not affect the significant figures. Rounding the surface area to 4 significant figures gives:
$$A\approx 311.3{\text{m}}^2$$

Comparing our calculated volume with the given options, the correct choice representing the volume is 373.7 m³.