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. 1,674.7 mm3
1,674.7 mm cubed

3,900 mm3
3,900 mm cubed

2,325.3 mm3
2,325.3 mm cubed

33.5 mm3
33.5 mm cubed

Answer 1

To determine how much space in the box is unused, we need to find the volume of the marbles and subtract it from the total volume of the container.

1. **Calculate the volume of one marble**:
The formula for the volume \( V \) of a sphere is given by:
\[
V = \frac{4}{3} \pi r^3
\]
where \( r \) is the radius of the marble. Given that the radius \( r \) is 2 mm and using \(\pi \approx 3.14\):

\[
V_{\text{one marble}} = \frac{4}{3} \times 3.14 \times (2^3)
\]
\[
= \frac{4}{3} \times 3.14 \times 8
\]
\[
= \frac{32}{3} \times 3.14
\]
\[
\approx 33.51 \, \text{mm}^3 \, \text{(approximately)}
\]

2. **Calculate the total volume of 50 marbles**:
\[
V_{\text{50 marbles}} = 50 \times V_{\text{one marble}} = 50 \times 33.51
\]
\[
\approx 1675.5 \, \text{mm}^3
\]

3. **Find the unused space in the container**:
\[
V_{\text{unused}} = V_{\text{container}} - V_{\text{50 marbles}}
\]
\[
= 4000 - 1675.5 \approx 2324.5 \, \text{mm}^3
\]

Rounding to the nearest tenth, the unused space is approximately:

\[
\boxed{2324.5 \, \text{mm}^3}
\]

From the options provided, the closest is **2,325.3 mm³**.