To find the unused space in the container, we first need to calculate the total volume occupied by the marbles and then subtract this from the total volume of the container.
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Calculate the volume of one marble:
The formula for the volume of a sphere is given by: \[ V = \frac{4}{3} \pi r^3 \] Where \( r \) is the radius. For one marble with a radius of 2 mm: \[ V = \frac{4}{3} \times 3.14 \times (2)^3 \] \[ V = \frac{4}{3} \times 3.14 \times 8 \] \[ V = \frac{4 \times 3.14 \times 8}{3} \] \[ V = \frac{100.48}{3} \approx 33.4933 \text{ mm}^3 \] -
Calculate the total volume occupied by 50 marbles:
Total volume for 50 marbles: \[ V_{total} = 50 \times 33.4933 \approx 1674.665 \text{ mm}^3 \] -
Calculate the unused space in the container:
The total volume of the container is given as 4,000 mm³. Therefore, the unused space: \[ V_{unused} = V_{container} - V_{total} \] \[ V_{unused} = 4000 - 1674.665 \approx 2325.335 \text{ mm}^3 \] -
Round the answer to the nearest tenth:
\[ V_{unused} \approx 2325.3 \text{ mm}^3 \]
The unused space in the box is 2,325.3 mm³. Thus, the correct response is 2,325.3 mm³.
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