Answer: $8893$

Explanation:1. Understand the goal: The question asks to convert a volume from cubic inches to cubic centimeters.

2. Identify the given information: Volume in cubic inches = $542.68 \text{ in}^3$. Conversion factor: $1 \text{ inch} = 2.54 \text{ centimeters}$.

3. Determine the conversion factor for volume: Since volume is a three-dimensional measure, we need to cube the linear conversion factor. If $1 \text{ inch} = 2.54 \text{ cm}$, then $(1 \text{ inch})^3 = (2.54 \text{ cm})^3$. So, $1 \text{ in}^3 = (2.54)^3 \text{ cm}^3$. Calculate $(2.54)^3$: $(2.54)^3 = 2.54 \times 2.54 \times 2.54 = 6.4516 \times 2.54 = 16.387064 \text{ cm}^3$. So, $1 \text{ in}^3 = 16.387064 \text{ cm}^3$.

4. Convert the given volume: Multiply the volume in cubic inches by the cubic conversion factor: Volume in $\text{cm}^3$ = Volume in $\text{in}^3 \times \dfrac{\text{cm}^3}{\text{in}^3}$ Volume in $\text{cm}^3 = 542.68 \text{ in}^3 \times 16.387064 \dfrac{\text{cm}^3}{\text{in}^3}$ Volume in $\text{cm}^3 = 542.68 \times 16.387064$ Volume in $\text{cm}^3 \approx 8892.5985...$

5. Round to the nearest whole number: The question asks for the volume to the nearest whole number. $8892.5985...$ rounded to the nearest whole number is 8893.

6. Conclusion: The volume of the firework's shell is approximately $8893 \text{ cm}^3$.