To determine how much space in the box is unused, we need to calculate the total volume occupied by the marbles and then subtract that from the volume of the container.
Step 1: Calculate the volume of a single marble.
The formula for the volume of a sphere is:
\[ V = \frac{4}{3} \pi r^3 \]
Where \( r \) is the radius of the sphere (marble). Given that each marble has a radius of 2 mm, we can substitute the values into the formula:
\[ V = \frac{4}{3} \times 3.14 \times (2)^3 \]
Calculating \( (2)^3 \):
\[ (2)^3 = 8 \]
Now substituting:
\[ V = \frac{4}{3} \times 3.14 \times 8 \]
Calculating:
\[ V = \frac{4 \times 3.14 \times 8}{3} = \frac{100.48}{3} \approx 33.4933 , \text{mm}^3 \]
Step 2: Calculate the total volume occupied by all 50 marbles.
Since there are 50 marbles:
\[ \text{Total Volume of Marbles} = 50 \times 33.4933 \approx 1674.665 , \text{mm}^3 \]
Step 3: Calculate the unused space in the box.
Now, we can subtract the total volume of the marbles from the volume of the container:
\[ \text{Unused Space} = \text{Volume of Container} - \text{Total Volume of Marbles} \]
Given that the volume of the container is 4,000 mm³:
\[ \text{Unused Space} = 4000 , \text{mm}^3 - 1674.665 , \text{mm}^3 \approx 2325.335 , \text{mm}^3 \]
Step 4: Round to the nearest tenth.
Rounding 2325.335 to the nearest tenth gives:
\[ \text{Unused Space} \approx 2325.3 , \text{mm}^3 \]
Final Answer:
The amount of unused space in the box is approximately 2,325.3 mm³. Thus, the correct answer is:
4. 2,325.3 mm³