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.

1 answer

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.)