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
3,900 mm3
3,900 mm cubed
1,674.7 mm3
1,674.7 mm cubed
2,325.3 mm3
2,325.3 mm cubed
33.5 mm3
To find out how much space is unused in the container, we first need to calculate the total volume occupied by the 50 marbles and then subtract that volume from the total volume of the container.
The volume \( V \) of a single sphere (marble) can be calculated using the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Where \( r \) is the radius and \( \pi \) is Pi (approximately 3.14).
Given that each marble has a radius of 2 mm, the volume of one marble is:
\[ V_{marble} = \frac{4}{3} \times 3.14 \times (2)^3 \]
\[ V_{marble} = \frac{4}{3} \times 3.14 \times 8 \]
\[ V_{marble} = \frac{4}{3} \times 25.12 \]
\[ V_{marble} = \frac{100.48}{3} \]
\[ V_{marble} = 33.49 \text{ mm}^3 \] (rounded to two decimal places)
Since there are 50 marbles, the total volume occupied by the marbles is:
\[ V_{total\ marbles} = 50 \times 33.49 \]
\[ V_{total\ marbles} = 1674.5 \text{ mm}^3 \] (rounded to one decimal place)
Now subtract this volume from the total volume of the container (4,000 mm³) to find the unused space:
\[ \text{Unused volume} = \text{Total container volume} - \text{Total marbles volume} \]
\[ \text{Unused volume} = 4000 \text{ mm}^3 - 1674.5 \text{ mm}^3 \]
\[ \text{Unused volume} = 2325.5 \text{ mm}^3 \]
Rounded to the nearest tenth, the unused space in the container is 2325.5 mm³. Therefore, the answer is:
2,325.3 mm³ or 2,325.3 mm cubed