The volume of a container measures 4,000 mm3 and contains 50 marbles. Each marble has a radius of 2 mm. Use 3.14 as an approximation for pi. How much space in the box is unused? Round your answer to the nearest tenth.(1 point) Responses 2,325.3 mm3 2,325.3 mm cubed 33.5 mm3 33.5 mm cubed 1,674.7 mm3 1,674.7 mm cubed 3,900 mm3
First, we calculate the volume of one marble. The formula for the volume of a sphere is \( \frac{4}{3} \pi r^3 \), where \( r \) is the radius of the sphere.
Given that the radius \( r \) of each marble is 2 mm and \( \pi \) is approximated as 3.14, the volume \( V \) of one marble is:
\( V = \frac{4}{3} \times 3.14 \times (2 \text{ mm})^3 \)
\( V = \frac{4}{3} \times 3.14 \times 8 \text{ mm}^3 \)
\( V = 4.18667 \times 8 \text{ mm}^3 \)
\( V \approx 33.5 \text{ mm}^3 \)
This is the approximate volume of one marble.
With 50 marbles in the container, the total volume occupied by the marbles is:
\( \text{Total volume of marbles} = 50 \times 33.5 \text{ mm}^3 \)
\( \text{Total volume of marbles} = 1675 \text{ mm}^3 \)
The container's volume is 4000 mm^3. To find the unused space, we subtract the total volume occupied by the marbles from the container volume:
\( \text{Unused space} = \text{Container volume} - \text{Total volume of marbles} \)
\( \text{Unused space} = 4000 \text{ mm}^3 - 1675 \text{ mm}^3 \)
\( \text{Unused space} = 2325 \text{ mm}^3 \)
Rounded to the nearest tenth, the unused space in the container is 2325 mm^3.
The correct answer is 2,325.3 mm^3 (or mm cubed).