To determine how many complete boxes can be wrapped with the roll of wrapping paper, we first need to calculate the area of the roll of wrapping paper and the area needed to wrap one box.
Step 1: Calculate the area of the roll of wrapping paper. The dimensions of the roll of wrapping paper are 25 in. × 405 in.
\[ \text{Area of the roll} = 25 \text{ in.} \times 405 \text{ in.} = 10125 \text{ in}^2 \]
Step 2: Calculate the surface area of one box. The dimensions of the box are 16 in. × 14 in. × 6 in.
The surface area \( A \) of a rectangular box can be calculated using the formula: \[ A = 2(lw + lh + wh) \] where \( l \), \( w \), and \( h \) are the length, width, and height of the box.
Substituting the dimensions of the box:
- \( l = 16 \) in.
- \( w = 14 \) in.
- \( h = 6 \) in.
\[ A = 2(16 \times 14 + 16 \times 6 + 14 \times 6) \] Calculating each term:
- \( 16 \times 14 = 224 \)
- \( 16 \times 6 = 96 \)
- \( 14 \times 6 = 84 \)
Now substitute back into the equation: \[ A = 2(224 + 96 + 84) \] Adding the values inside the parentheses: \[ 224 + 96 + 84 = 404 \] Calculating the total surface area: \[ A = 2 \times 404 = 808 \text{ in}^2 \]
Step 3: Calculate how many boxes can be wrapped with the wrapping paper. Now we divide the total area of the wrapping paper by the surface area of one box: \[ \text{Number of boxes} = \frac{\text{Area of the roll}}{\text{Surface area of one box}} = \frac{10125 \text{ in}^2}{808 \text{ in}^2} \]
Calculating the number of boxes: \[ \text{Number of boxes} \approx 12.5 \]
Since we can only wrap complete boxes, we take the floor of this value: \[ \text{Number of complete boxes} = 12 \]
Therefore, with one roll of wrapping paper, you can wrap 12 complete boxes.