To find out how much space is unused 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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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. Given that the radius of each marble is 2 mm:
\[ V = \frac{4}{3} \times 3.14 \times (2)^3 \]
Calculate \( (2)^3 \):
\[ (2)^3 = 8 \]
Now plug that value back into the formula:
\[ V = \frac{4}{3} \times 3.14 \times 8 \]
Calculate \( \frac{4}{3} \times 8 \):
\[ \frac{4 \times 8}{3} = \frac{32}{3} \approx 10.67 \]
Now, multiply by 3.14:
\[ V \approx 10.67 \times 3.14 \approx 33.478 \]
So the volume of one marble is approximately \( 33.5 , \text{mm}^3 \) (when rounded to the nearest tenth).
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Total volume occupied by all 50 marbles:
\[ \text{Total Volume of Marbles} = 50 \times 33.5 \approx 1675 , \text{mm}^3 \]
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Calculating unused space in the container:
The total volume of the container is 4,000 mm³. Thus, the unused space is calculated as follows:
\[ \text{Unused Space} = \text{Total Volume of Container} - \text{Total Volume of Marbles} \]
\[ \text{Unused Space} = 4000 , \text{mm}^3 - 1675 , \text{mm}^3 = 2325 , \text{mm}^3 \]
Therefore, the unused space in the box is approximately 2,325.3 mm³ when rounded to the nearest tenth.
The correct response is:
2,325.3 mm³.
1 answer