Volume of Spheres Quick Check

Question

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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

1,674.7 mm3
1,674.7 mm cubed

3,900 mm3
3,900 mm cubed

33.5 mm3
33.5 mm cubed

2,325.3 mm3

Answer 1

To find the unused space in the box, we need to calculate the total volume of the marbles and then subtract that from the volume of the container.

1. Calculate the volume of a single marble: The formula for the volume \( V \) of a sphere is: \[ V = \frac{4}{3} \pi r^3 \] Given that the radius \( r = 2 \) mm and using \( \pi \approx 3.14 \): \[ V = \frac{4}{3} \times 3.14 \times (2)^3 \] First calculate \( (2)^3 = 8 \): \[ V = \frac{4}{3} \times 3.14 \times 8 \] \[ V = \frac{32}{3} \times 3.14 \approx 33.49 , \text{mm}^3 \quad (\text{approximately}) \]

2. Calculate the total volume of all 50 marbles: \[ \text{Total volume of marbles} = 50 \times 33.49 \approx 1,674.5 , \text{mm}^3 \]

3. Calculate the unused space in the box: \[ \text{Unused space} = \text{Volume of the container} - \text{Total volume of marbles} \] \[ \text{Unused space} = 4,000 , \text{mm}^3 - 1,674.5 , \text{mm}^3 \approx 2,325.5 , \text{mm}^3 \]

Rounding to the nearest tenth gives us: \[ \text{Unused space} \approx 2,325.5 , \text{mm}^3 \]

From the given choices, the answer that is closest to this calculation is: 2,325.3 mm³.