To determine how many complete boxes can be wrapped with one roll of wrapping paper, we first need to calculate the area of both the roll of 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 of wrapping paper are:
- Width = 25 inches
- Length = 405 inches
The area \( A \) of the roll is given by the formula: \[ A_{roll} = \text{Width} \times \text{Length} \] \[ A_{roll} = 25 , in. \times 405 , in. = 10125 , in.^2 \]
Step 2: Calculate the surface area of one box.
The dimensions of the box are:
- Length = 16 inches
- Width = 14 inches
- Height = 6 inches
The surface area \( A \) of a rectangular box can be calculated with the formula: \[ A_{box} = 2(lw + lh + wh) \] Where:
- \( l \) = length
- \( w \) = width
- \( h \) = height
Substituting in the dimensions: \[ A_{box} = 2(16 \times 14 + 16 \times 6 + 14 \times 6) \]
Calculating each term:
- \( 16 \times 14 = 224 \)
- \( 16 \times 6 = 96 \)
- \( 14 \times 6 = 84 \)
Now, summing these areas: \[ 224 + 96 + 84 = 404 \]
And multiplying by 2 to get the total surface area: \[ A_{box} = 2 \times 404 = 808 , in.^2 \]
Step 3: Calculate the number of complete boxes that can be wrapped.
Now that we have the area of the roll of wrapping paper and the surface area of one box, we can find out how many complete boxes can be wrapped by dividing the area of the roll by the area of one box: \[ \text{Number of boxes} = \frac{A_{roll}}{A_{box}} = \frac{10125 , in.^2}{808 , in.^2} \]
Calculating this gives: \[ \text{Number of boxes} \approx 12.5 \]
Since we can only wrap whole boxes, we take the integer part of this number: \[ \text{Complete boxes} = 12 \]
Thus, 12 complete boxes can be wrapped with one roll of wrapping paper.