To calculate how many 8 mm diameter spheres can fit into the rectangular box, we need to consider how the spheres will be arranged inside the box.

Steps:

1. Volume-based approximation:

   • Calculate the volume of the box.
   • Calculate the volume of a single sphere.
   • Estimate the number of spheres by dividing the box's volume by the sphere's volume.

2. Packing efficiency:

   • Spheres do not pack perfectly due to empty spaces between them. For efficient packing, we use the closest packing arrangement, such as the hexagonal closest packing or cubic packing. In such arrangements, the spheres occupy approximately 74% of the box volume.

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Step 1: Volume Calculation

• Volume of the rectangular box = length × width × height $V_{\text{box}} = 105 \, \text{mm} \times 70 \, \text{mm} \times 30 \, \text{mm} = 220,500 \, \text{mm}^3$

• Volume of one sphere = $\frac{4}{3} \pi r^3$ where $r = \frac{8}{2} = 4 \, \text{mm}$. $V_{\text{sphere}} = \frac{4}{3} \pi (4)^3 = \frac{4}{3} \pi \times 64 = 268.08 \, \text{mm}^3$

Step 2: Packing Efficiency

For closest packing (hexagonal), approximately 74% of the box's volume is filled.

$\text{Effective volume for spheres} = 0.74 \times 220,500 \, \text{mm}^3 = 163,170 \, \text{mm}^3$

Step 3: Number of Spheres

$\text{Number of spheres} = \frac{163,170}{268.08} \approx 609$

Final Answer:

Approximately 609 spheres of 8 mm diameter can fit into the rectangular box.

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Would you like further details on how packing efficiency is calculated or any other aspect of this solution?

Here are 5 related questions you might find interesting:

1. How does sphere packing change with different shapes of containers?
2. What is the difference between cubic and hexagonal packing?
3. Can the packing efficiency be higher for irregular shapes?
4. How would the calculation change for non-uniform spheres?
5. How do you calculate the surface area occupied by the spheres?

Tip: Packing problems often benefit from simulation techniques when the arrangement of objects is complex!