Question

Surface Area of Right Rectangular Prisms Practice
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5 of 55 of 5 Items
Question
A roll of wrapping paper measures 25 in.×405 in.
The boxes being wrapped are 16 in.×14 in.×6 in.
How many complete boxes can be wrapped with one roll of wrapping paper?(1 point)

Answer 1

To determine how many complete boxes can be wrapped with one roll of wrapping paper, we'll first need to calculate the surface area of both the roll of wrapping paper and the boxes.

1. **Calculate the Surface Area of the Box:**
The formula for the surface area \(SA\) of a rectangular prism (box) is:
\[
SA = 2(lw + lh + wh)
\]
where \(l\), \(w\), and \(h\) are the length, width, and height of the box, respectively.

Given the dimensions of the box:
- Length \(l = 16 \, \text{in}\)
- Width \(w = 14 \, \text{in}\)
- Height \(h = 6 \, \text{in}\)

Now, substitute the values:
\[
SA = 2(16 \times 14 + 16 \times 6 + 14 \times 6)
\]

**Calculate each term:**
- \(16 \times 14 = 224\)
- \(16 \times 6 = 96\)
- \(14 \times 6 = 84\)

Now plug in these values:
\[
SA = 2(224 + 96 + 84) = 2(404) = 808 \, \text{sq in}
\]

2. **Calculate the Surface Area of the Roll of Wrapping Paper:**
The dimensions of the roll are:
- Width = 25 in
- Length = 405 in

Since the wrap is essentially a rectangle, we can view the surface area as a flat rectangle, which can be calculated as:
\[
SA_{paper} = \text{Width} \times \text{Length} = 25 \, \text{in} \times 405 \, \text{in}
\]
\[
SA_{paper} = 10125 \, \text{sq in}
\]

3. **Determine How Many Boxes Can Be Wrapped:**
Now, divide the total surface area of the wrapping paper by the surface area of one box:
\[
\text{Number of boxes} = \frac{SA_{paper}}{SA_{box}} = \frac{10125 \, \text{sq in}}{808 \, \text{sq in}}
\]
\[
\text{Number of boxes} \approx 12.5
\]

Since we can only wrap complete boxes, we round down to the nearest whole number:
\[
\text{Complete boxes wrapped} = 12
\]

Therefore, **the answer is 12 complete boxes.**