To determine how many complete boxes can be wrapped with one roll of wrapping paper, we need to first calculate the area of both the wrapping paper and the surface area of one box.
Step 1: Calculate the area of the roll of wrapping paper.
The dimensions of the roll are 25 inches by 405 inches. The area \( A \) of the wrapping paper can be calculated as:
\[ A_{\text{paper}} = \text{width} \times \text{length} = 25 , \text{in} \times 405 , \text{in} = 10125 , \text{in}^2 \]
Step 2: Calculate the surface area of one box.
The dimensions of the box are 16 inches, 14 inches, and 6 inches. The surface area \( A \) of a rectangular box is calculated using the formula:
\[ A_{\text{box}} = 2(lw + lh + wh) \]
where \( l \), \( w \), and \( h \) are the length, width, and height of the box.
Using the box dimensions:
- \( l = 16 , \text{in} \)
- \( w = 14 , \text{in} \)
- \( h = 6 , \text{in} \)
We can substitute these values:
\[ A_{\text{box}} = 2(16 \times 14 + 16 \times 6 + 14 \times 6) \]
Calculating each term inside the parentheses:
- \( 16 \times 14 = 224 \)
- \( 16 \times 6 = 96 \)
- \( 14 \times 6 = 84 \)
Now sum these values:
\[ 224 + 96 + 84 = 404 \]
So the surface area of the box is:
\[ A_{\text{box}} = 2 \times 404 = 808 , \text{in}^2 \]
Step 3: Calculate how many boxes can be wrapped.
Now, we divide the total area of the wrapping paper by the surface area of one box:
\[ \text{Number of boxes} = \frac{A_{\text{paper}}}{A_{\text{box}}} = \frac{10125 , \text{in}^2}{808 , \text{in}^2} \approx 12.5 \]
Since we can only wrap complete boxes, we take the integer part:
\[ \text{Complete boxes} = 12 \]
Thus, 12 complete boxes can be wrapped with one roll of wrapping paper.