A company packages puzzles in boxes that are 6 inches by 4 inches by 2 inches. The puzzle boxes are packed into a shipping crate. The shipping crate is cube with 12 - inches.

Question

What is the greatest number of puzzle boxes that can be packed into a shipping crate?

Answer 1

To determine the greatest number of puzzle boxes that can fit into a shipping crate, we first need to calculate the volume of both the puzzle boxes and the shipping crate.

1. Volume of a puzzle box: The dimensions of the puzzle box are 6 inches by 4 inches by 2 inches. \[ \text{Volume of a puzzle box} = \text{length} \times \text{width} \times \text{height} = 6 , \text{inches} \times 4 , \text{inches} \times 2 , \text{inches} = 48 , \text{cubic inches} \]

2. Volume of the shipping crate: The shipping crate is a cube with each side measuring 12 inches. \[ \text{Volume of the shipping crate} = \text{side}^3 = 12 , \text{inches} \times 12 , \text{inches} \times 12 , \text{inches} = 1728 , \text{cubic inches} \]

3. Determine how many puzzle boxes can fit: To find out how many puzzle boxes can fit into the volume of the shipping crate, we can divide the volume of the shipping crate by the volume of one puzzle box. \[ \text{Number of puzzle boxes} = \frac{\text{Volume of shipping crate}}{\text{Volume of puzzle box}} = \frac{1728 , \text{cubic inches}}{48 , \text{cubic inches}} = 36 \]

4. Check the arrangement: It’s important to also check the dimensions of the boxes to ensure they can be arranged within the crate dimensions. The shipping crate dimensions are 12 inches on each side.

   • Fit along the 12-inch side:
     • Length-wise (6 inches): \( \frac{12}{6} = 2 \)
     • Width-wise (4 inches): \( \frac{12}{4} = 3 \)
     • Height-wise (2 inches): \( \frac{12}{2} = 6 \)

   Thus, the total number of puzzle boxes that can be fitted in the crate is: \[ 2 , (\text{length}) \times 3 , (\text{width}) \times 6 , (\text{height}) = 36 \]

Therefore, the greatest number of puzzle boxes that can be packed into the shipping crate is \( \boxed{36} \).

Answer 2

are you sure

Answer 3

Let's double-check the calculations and the arrangement to ensure accuracy.

1. Volume Calculations

Volume of a puzzle box:

• Dimensions: 6 inches (length) × 4 inches (width) × 2 inches (height)
• \[ \text{Volume of a puzzle box} = 6 \times 4 \times 2 = 48 \text{ cubic inches} \]

Volume of the shipping crate:

• The shipping crate is a cube with each side measuring 12 inches.
• \[ \text{Volume of the shipping crate} = 12^3 = 12 \times 12 \times 12 = 1728 \text{ cubic inches} \]

2. Number of Puzzle Boxes in Terms of Volume

• To find out how many of the puzzle boxes can fit in the shipping crate based on volume:

\[ \text{Number of puzzle boxes (based on volume)} = \frac{1728}{48} = 36 \]

3. Packing Arrangement

Now, we need to consider how these boxes can be arranged within the dimensions of the crate to confirm that the total number calculated is feasible.

Dimensions of the shipping crate: 12 inches × 12 inches × 12 inches.

Fitting puzzle boxes in the crate:

• Along 12 inches (length):
  • 6 inches long: \(\frac{12}{6} = 2\) boxes
• Along 12 inches (width):
  • 4 inches wide: \(\frac{12}{4} = 3\) boxes
• Along 12 inches (height):
  • 2 inches high: \(\frac{12}{2} = 6\) boxes

Thus, the arrangement allows us to fit: \[ 2 \times 3 \times 6 = 36 \text{ boxes} \]

Conclusion

Both methods of calculation—volume-based and arrangement-based—confirm that the maximum number of puzzle boxes that can be packed into the shipping crate is indeed \( \boxed{36} \).