To find the volume of the emerald, we start with the formula for density:
[tex]\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \][/tex]
From this, we can rearrange the formula to solve for volume:
[tex]\[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} \][/tex]
Given the mass of the emerald is 378.24 grams and the density is [tex]\(2.76 \, \frac{\text{grams}}{\text{cm}^3}\)[/tex], we substitute these values into the formula:
[tex]\[ \text{Volume} = \frac{378.24 \, \text{grams}}{2.76 \, \frac{\text{grams}}{\text{cm}^3}} \][/tex]
Performing the division:
[tex]\[ \text{Volume} \approx 137.0434782608696 \, \text{cm}^3 \][/tex]
To match the problem requirement to round the volume to the nearest hundredth:
[tex]\[ \text{Volume} \approx 137.04 \, \text{cm}^3 \][/tex]
Therefore, the volume of the emerald, rounded to the nearest hundredth, is:
137.04
