First, we need to calculate the volume of one marble. The formula for the volume \( V \) of a sphere is:
\[ V = \frac{4}{3} \pi r^3 \]
Here, the radius \( r \) of each marble is 2 mm. Using \( \pi \approx 3.14 \), we can substitute the values into the formula:
\[ V = \frac{4}{3} \times 3.14 \times (2)^3 \]
Calculating \( (2)^3 \):
\[ (2)^3 = 8 \]
Now substitute back into the volume formula:
\[ V = \frac{4}{3} \times 3.14 \times 8 \]
Calculating \( \frac{4}{3} \times 8 \):
\[ \frac{4 \times 8}{3} = \frac{32}{3} \approx 10.6667 \]
Now multiplying by 3.14:
\[ V \approx 10.6667 \times 3.14 \approx 33.478 \]
So, the volume of one marble is approximately \( 33.5 , \text{mm}^3 \) when rounded to one decimal place.
Next, we calculate the total volume occupied by the 50 marbles:
\[ \text{Total volume of marbles} = 50 \times 33.5 \approx 1675 , \text{mm}^3 \]
Now, to find the unused space in the container, we subtract the volume occupied by the marbles from the total volume of the container:
\[ \text{Unused space} = \text{Volume of container} - \text{Total volume of marbles} \] \[ \text{Unused space} = 4000 , \text{mm}^3 - 1675 , \text{mm}^3 = 2325 , \text{mm}^3 \]
Rounding this to the nearest tenth gives:
\[ \text{Unused space} \approx 2325.0 , \text{mm}^3 \]
Thus, the unused space in the box is:
2,325.3 mm³ (correct option).
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