To determine how many oranges can fit into the box, we can break down the problem step by step:

Step 1: Calculate the volume of the box

The volume $V_{\text{box}}$ of a rectangular box is given by the formula: $V_{\text{box}} = \text{length} \times \text{width} \times \text{height}$ Substituting the given dimensions of the box (11 inches, 21 inches, 10 inches): $V_{\text{box}} = 11 \times 21 \times 10 = 2310 \, \text{cubic inches}$

Step 2: Calculate the volume of one orange

The volume $V_{\text{orange}}$ of a sphere is given by the formula: $V_{\text{orange}} = \frac{4}{3} \pi r^3$ where $r$ is the radius of the sphere. Since the diameter of the orange is 2.8 inches, the radius $r = \frac{2.8}{2} = 1.4$ inches: $V_{\text{orange}} = \frac{4}{3} \pi (1.4)^3 \approx \frac{4}{3} \pi (2.744) \approx 11.52 \, \text{cubic inches}$

Step 3: Calculate the total number of oranges based on volume

If the spheres were perfectly packed, the total number of oranges $N$ could be estimated as: $N_{\text{perfect}} = \frac{V_{\text{box}}}{V_{\text{orange}}} = \frac{2310}{11.52} \approx 200.52$ So, under perfect packing conditions, around 200 oranges would fit.

Step 4: Adjust for random packing efficiency

Since random packing of spheres typically takes up around 190% of the volume of the spheres (due to inefficient use of space), we need to reduce the number of oranges accordingly: $N_{\text{actual}} = \frac{N_{\text{perfect}}}{1.9} = \frac{200.52}{1.9} \approx 105.54$

Conclusion:

Rounding to the nearest whole number, approximately 106 oranges should fit into the box.

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Would you like more details on any specific part of this solution, or do you have any questions?

Here are 5 questions to explore this topic further:

1. How does the packing efficiency affect the overall number of oranges?
2. What would happen if the orange size changes to 3 inches in diameter?
3. How would the result change with more efficient (hexagonal close) packing instead of random packing?
4. What if the box had curved edges or was a cylindrical shape?
5. How does changing the material properties of oranges (e.g., squishiness) affect packing?

Tip: Remember that efficient packing methods like face-centered cubic (FCC) or hexagonal close packing (HCP) can fit more spheres than random packing!