To find out how much space in the box is unused, we need to find the 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 \( V \) of a sphere is: \[ V = \frac{4}{3} \pi r^3 \] where \( r \) is the radius.
Given that the radius \( r = 2 \) mm, using \( \pi \approx 3.14 \): \[ V = \frac{4}{3} \times 3.14 \times (2)^3 \] \[ V = \frac{4}{3} \times 3.14 \times 8 \] \[ V = \frac{4 \times 3.14 \times 8}{3} \] \[ V = \frac{100.48}{3} \approx 33.49 \text{ mm}^3 \quad (\text{rounded to 2 decimal places}) \]
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Calculate the total volume occupied by 50 marbles: \[ \text{Total volume of marbles} = 50 \times 33.49 \approx 1674.5 \text{ mm}^3 \]
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Calculate the unused space in the container: The total volume of the container is 4,000 mm³. \[ \text{Unused space} = \text{Total volume} - \text{Volume occupied by marbles} \] \[ \text{Unused space} = 4000 - 1674.5 = 2325.5 \text{ mm}^3 \]
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Round to the nearest tenth: The unused space rounded to the nearest tenth is: \[ 2325.5 \text{ mm}^3 \approx 2325.3 \text{ mm}^3 \]
Thus, the answer is 2,325.3 mm³.
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