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.
First, let's calculate the volume of a single marble. The formula to calculate the volume of a sphere (the marble) is:
\[ V = \frac{4}{3} \pi r^3 \]
Where \( V \) is the volume, \( \pi \) is the constant pi (approximately 3.14 here), and \( r \) is the radius.
Given that the radius \( r \) of each marble is 2 mm, the volume \( V \) of one marble would be:
\[ V = \frac{4}{3} \times 3.14 \times (2)^3 \]
\[ V = \frac{4}{3} \times 3.14 \times 8 \]
\[ V = \frac{4}{3} \times 25.12 \]
\[ V = 33.493333... \text{ mm}^3 \]
Now, we need to calculate the volume of 50 marbles:
\[ V_{total} = 50 \times V_{single} \]
\[ V_{total} = 50 \times 33.493333... \]
\[ V_{total} = 1674.666666... \text{ mm}^3 \]
This is the total volume occupied by 50 marbles. Now we will subtract this from the total volume of the container to find the unused space:
\[ V_{unused} = V_{container} - V_{total} \]
\[ V_{unused} = 4000 - 1674.666666... \]
\[ V_{unused} = 2325.333333... \text{ mm}^3 \]
To round this to the nearest tenth:
\[ V_{unused} \approx 2325.3 \text{ mm}^3 \]
So, approximately 2325.3 mm³ of the container's space is unused.