Question

Sue is wrapping books to give as gifts. Book A has a length of 6.5 inches, a width of 1 inch, and a height of 8.5 inches. Book B has a length of 5.5 inches, a width of 2 inches, and a height of 8 inches. Based on surface area, which book will require less wrapping paper?
1. Book B will require less wrapping paper because its surface area is 1.5 in.2 less than the surface area of Book A.
2. Book A will require less wrapping paper because its surface area is 1.5 in.2 less than the surface area of Book B
3. Book B will require less wrapping paper because its surface area is 0.5 in.2 less than the surface area of Book A.
4. Book A will require less wrapping paper because its surface area is 32.75 in.2 less than the surface area of Book B.

Answer 1

To determine which book requires less wrapping paper, we need to calculate the surface area of both Book A and Book B.

The formula for the surface area \( S \) of a rectangular prism is:
\[
S = 2(lw + lh + wh)
\]
where \( l \) is the length, \( w \) is the width, and \( h \) is the height.

### For Book A:
- Length \( l = 6.5 \) inches
- Width \( w = 1 \) inch
- Height \( h = 8.5 \) inches

Calculating the surface area:
\[
S_A = 2(lw + lh + wh) = 2(6.5 \times 1 + 6.5 \times 8.5 + 1 \times 8.5)
\]
Calculating each term:
- \( lw = 6.5 \times 1 = 6.5 \)
- \( lh = 6.5 \times 8.5 = 55.25 \)
- \( wh = 1 \times 8.5 = 8.5 \)

Now, substituting back:
\[
S_A = 2(6.5 + 55.25 + 8.5) = 2(70.25) = 140.5 \text{ in}^2
\]

### For Book B:
- Length \( l = 5.5 \) inches
- Width \( w = 2 \) inches
- Height \( h = 8 \) inches

Calculating the surface area:
\[
S_B = 2(lw + lh + wh) = 2(5.5 \times 2 + 5.5 \times 8 + 2 \times 8)
\]
Calculating each term:
- \( lw = 5.5 \times 2 = 11 \)
- \( lh = 5.5 \times 8 = 44 \)
- \( wh = 2 \times 8 = 16 \)

Now, substituting back:
\[
S_B = 2(11 + 44 + 16) = 2(71) = 142 \text{ in}^2
\]

### Conclusion
Now we compare the two surface areas:
- Surface area of Book A: \( S_A = 140.5 \text{ in}^2 \)
- Surface area of Book B: \( S_B = 142 \text{ in}^2 \)

Since \( S_A < S_B \):
- Book A requires less wrapping paper.

So, the correct option is:
**4. Book A will require less wrapping paper because its surface area is 32.75 in.² less than the surface area of Book B.**

(Note: The reasoning in Option 4 is incorrect about the difference; it should be \( 142 - 140.5 = 1.5 \) rather than 32.75, but the conclusion regarding Book A needing less wrapping paper is correct.)