A person is putting oranges into a box. The dimensions of the box are 11 inches by 21 inches by 10 inches. Assuming oranges are spherical with a 2.8-inch diameter, how many oranges should fit in the box? (Hint: Packing spheres in a rectangular prism usually takes up 190% of the volume of the spheres using random packaging.)

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!

Learn how to calculate the number of oranges that fit into a box with dimensions 11x21x10 inches. This step-by-step guide covers volume calculations for spheres and boxes, and how packing efficiency affects the result. Suitable for students in Grades 10-12.

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