To find the size of the container that will fit the 10 cartons, we need to determine the total volume and then consider the most efficient arrangement.

Step 1: Calculate the volume of one carton.

Each carton has the following dimensions:

• Length: 7 inches
• Width: 7 inches
• Height: 7.5 inches

The volume of one carton is calculated by multiplying these dimensions:

$\text{Volume of one carton} = 7 \times 7 \times 7.5 = 367.5 \, \text{cubic inches}$

Step 2: Calculate the total volume of 10 cartons.

Since you have 10 cartons:

$\text{Total volume} = 10 \times 367.5 = 3675 \, \text{cubic inches}$

Step 3: Estimate the container size.

Now, we can find the most efficient container size by considering how the cartons can be arranged in the container. We’ll aim to place them in rows, columns, and layers, keeping the dimensions as compact as possible.

Option 1: Arranging them in a grid (2 rows x 5 columns)

• Length: 5 cartons along the length (5 × 7 inches = 35 inches)
• Width: 2 cartons along the width (2 × 7 inches = 14 inches)
• Height: 1 layer with a height of 7.5 inches

Thus, the container dimensions would be approximately:

• Length: 35 inches
• Width: 14 inches
• Height: 7.5 inches

Option 2: Arranging them differently (1 row x 10 columns)

• Length: 10 cartons along the length (10 × 7 inches = 70 inches)
• Width: 1 carton along the width (7 inches)
• Height: 1 layer with a height of 7.5 inches

In this case, the container dimensions would be:

• Length: 70 inches
• Width: 7 inches
• Height: 7.5 inches

Option 3: A more compact arrangement (2 layers)

If we stack two layers, then:

• Length: 5 cartons along the length (5 × 7 inches = 35 inches)
• Width: 2 cartons along the width (2 × 7 inches = 14 inches)
• Height: 2 layers (2 × 7.5 inches = 15 inches)

So the container dimensions would be:

• Length: 35 inches
• Width: 14 inches
• Height: 15 inches

Conclusion:

The most compact solution would be Option 3:

• Length: 35 inches
• Width: 14 inches
• Height: 15 inches

Would you like to explore other options or need further details?

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Here are 5 related questions for further understanding:

1. How do you calculate the most efficient packing for irregularly shaped objects?
2. What is the difference between volume and packing density?
3. How does container shape impact shipping costs?
4. What are the best practices for stacking boxes to minimize wasted space?
5. How do you calculate the optimal number of layers in a container?

Tip: Always consider different arrangements when packing to minimize wasted space—sometimes rotating or rearranging the items can result in a smaller container.