To find the unused space in the container, we first need to calculate the total volume occupied by the marbles and then subtract that from the total volume of the container.
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Calculate the volume of a single marble: The formula for the volume of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.
Given the radius of each marble is 2 mm, we can substitute:
\[ V = \frac{4}{3} \times 3.14 \times (2)^3 \] \[ V = \frac{4}{3} \times 3.14 \times 8 \] \[ V = \frac{32 \times 3.14}{3} \] \[ V = \frac{100.48}{3} \approx 33.49 \text{ mm}^3 \]
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Calculate the total volume occupied by 50 marbles:
\[ V_{total_marbles} = 50 \times V = 50 \times 33.49 \approx 1674.7 \text{ mm}^3 \]
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Calculate the unused space in the container:
The total volume of the container is given as 4,000 mm³.
\[ \text{Unused space} = V_{container} - V_{total_marbles} \] \[ \text{Unused space} = 4000 - 1674.7 \approx 2325.3 \text{ mm}^3 \]
So the unused space in the box is approximately 2325.3 mm³.
The correct answer from the options provided is 2,325.3 mm3.