To find the unused space in the box, we first need to calculate the total volume occupied by the 50 marbles and then subtract that from the total volume of the container.
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Calculate the volume of one marble: The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Where \( r \) is the radius. Given that the radius is 2 mm, we can substitute this into the formula:
\[ V_{\text{marble}} = \frac{4}{3} \times 3.14 \times (2)^3 \] \[ V_{\text{marble}} = \frac{4}{3} \times 3.14 \times 8 \] \[ V_{\text{marble}} = \frac{4}{3} \times 25.12 \] \[ V_{\text{marble}} = \frac{100.48}{3} \approx 33.49 , \text{mm}^3 \]
So, the volume of one marble is approximately 33.5 mm³.
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Calculate the total volume occupied by 50 marbles:
\[ V_{\text{total marbles}} = 50 \times V_{\text{marble}} = 50 \times 33.5 \approx 1675 , \text{mm}^3 \]
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Calculate the unused space in the container:
The total volume of the container is given as 4000 mm³. Therefore, the unused space is:
\[ V_{\text{unused}} = V_{\text{container}} - V_{\text{total marbles}} = 4000 - 1675 \approx 2325 , \text{mm}^3 \]
Finally, rounding this to the nearest tenth, the unused space is approximately:
\[ \boxed{2325.0 , \text{mm}^3} \]
So, the correct answer from the provided responses is 2,325.3 mm³.
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