To determine how much space in the box is unused, we first need to calculate the total volume occupied by the 50 marbles and then subtract that from the volume of the container.
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Calculate the volume of one marble using the formula for the volume of a sphere:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the marble.
Given that the radius \( r = 2 \) mm and using \( \pi \approx 3.14 \):
\[ V = \frac{4}{3} \times 3.14 \times (2^3) \]
Calculate \( 2^3 = 8 \):
\[ V = \frac{4}{3} \times 3.14 \times 8 \]
Now calculate \( \frac{4}{3} \times 8 = \frac{32}{3} \approx 10.67 \).
Now multiply by \( 3.14 \):
\[ V \approx 10.67 \times 3.14 \approx 33.5 \text{ mm}^3 \]
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Calculate the total volume for 50 marbles:
\[ \text{Total volume of 50 marbles} = 50 \times 33.5 \approx 1675 \text{ mm}^3 \]
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Subtract the total volume of the marbles from the volume of the container:
Given the volume of the container is 4,000 mm³:
\[ \text{Unused space} = \text{Volume of container} - \text{Total volume of 50 marbles} \]
\[ \text{Unused space} = 4000 \text{ mm}^3 - 1675 \text{ mm}^3 \approx 2325 \text{ mm}^3 \]
Rounding to the nearest tenth, the unused space in the box is:
2,325.3 mm³
Thus, the correct answer is:
2,325.3 mm³
1 answer