Question

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 33.5 mm3 33.5 mm cubed 3,900 mm3 3,900 mm cubed 2,325.3 mm3 2,325.3 mm cubed 1,674.7 mm3

Answer 1

To find the unused space 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.

1. **Calculate the volume of a single marble:**
The formula for the volume of a sphere is given by:

\[
V = \frac{4}{3} \pi r^3
\]

where \( r \) is the radius of the sphere.

Given the radius of each marble is 2 mm, we can substitute:

\[
V = \frac{4}{3} \times 3.14 \times (2)^3
\]
\[
V = \frac{4}{3} \times 3.14 \times 8
\]
\[
V = \frac{32 \times 3.14}{3}
\]
\[
V = \frac{100.48}{3} \approx 33.49 \text{ mm}^3
\]

2. **Calculate the total volume occupied by 50 marbles:**

\[
V_{total\_marbles} = 50 \times V = 50 \times 33.49 \approx 1674.7 \text{ mm}^3
\]

3. **Calculate the unused space in the container:**

The total volume of the container is given as 4,000 mm³.

\[
\text{Unused space} = V_{container} - V_{total\_marbles}
\]
\[
\text{Unused space} = 4000 - 1674.7 \approx 2325.3 \text{ mm}^3
\]

So the unused space in the box is approximately **2325.3 mm³**.

The correct answer from the options provided is **2,325.3 mm3**.