To determine how many complete boxes can be wrapped with one roll of wrapping paper, we first calculate the surface area of one box and the surface area of the roll of wrapping paper.
Step 1: Calculate Surface Area of One Box
The box has dimensions of 16 in. × 14 in. × 6 in.
The surface area \( S \) of a rectangular box is calculated using the formula:
\[ S = 2(lw + lh + wh) \]
where:
- \( l \) = length
- \( w \) = width
- \( h \) = height
Plugging in the values:
- \( l = 16 \) in
- \( w = 14 \) in
- \( h = 6 \) in
Calculating:
\[ S = 2(16 \times 14 + 16 \times 6 + 14 \times 6) \] \[ = 2(224 + 96 + 84) \] \[ = 2(404) \] \[ = 808 \text{ in}^2 \]
Step 2: Calculate Surface Area of the Roll of Wrapping Paper
The roll of wrapping paper has dimensions of 25 in. × 405 in.
The surface area \( A \) of the roll of wrapping paper is simply its length and width:
\[ A = 25 \times 405 \]
Calculating:
\[ A = 10125 \text{ in}^2 \]
Step 3: Determine How Many Boxes Can Be Wrapped
Next, we divide the surface area of the roll of wrapping paper by the surface area of one box:
\[ \text{Number of boxes} = \frac{\text{Area of roll}}{\text{Area of one box}} = \frac{10125}{808} \]
Calculating:
\[ \text{Number of boxes} \approx 12.5 \]
Since we can only wrap whole boxes, we take the whole number part.
Thus, the number of complete boxes that can be wrapped with one roll of wrapping paper is:
\[ \boxed{12} \]