To determine how many complete boxes can be wrapped with one roll of wrapping paper, we first need to calculate the surface area of both the wrapping paper and the boxes.
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Calculate the surface area of the box: A box has a surface area given by the formula: \[ \text{Surface Area} = 2(lw + lh + wh) \] where \( l \) is the length, \( w \) is the width, and \( h \) is the height of the box.
For the given box dimensions:
- Length \( l = 16 \) in
- Width \( w = 14 \) in
- Height \( h = 6 \) in
Now, substituting in the values: \[ \text{Surface Area} = 2(16 \cdot 14 + 16 \cdot 6 + 14 \cdot 6) \]
- \( 16 \cdot 14 = 224 \)
- \( 16 \cdot 6 = 96 \)
- \( 14 \cdot 6 = 84 \)
Adding those: \[ 224 + 96 + 84 = 404 \] So the surface area is: \[ \text{Surface Area} = 2 \cdot 404 = 808 \text{ square inches} \]
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Calculate the surface area of the wrapping paper roll: The roll of wrapping paper is 25 in. wide and 405 in. long. Therefore, the total surface area is: \[ \text{Surface Area of the Wrapping Paper} = \text{width} \times \text{length} = 25 \text{ in} \times 405 \text{ in} = 10125 \text{ square inches} \]
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Determine how many boxes can be wrapped: Now we can find out how many complete boxes can be wrapped by dividing the surface area of the wrapping paper by the surface area of a single box: \[ \text{Number of boxes} = \frac{\text{Surface Area of the Wrapping Paper}}{\text{Surface Area of One Box}} = \frac{10125}{808} \]
Calculating that gives: \[ \text{Number of boxes} \approx 12.5 \] Since we can only wrap complete boxes, we can wrap 12 complete boxes with the roll of wrapping paper.
The final answer is: \[ \boxed{12} \]