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It takes 1/8 of a roll of wrapping paper to completely cover all 6 sides of a small box that is shaped like a rectangular prism. The box has a volume of 10 cubic inches. Suppose the dimensions of the box are tripled. How many rolls of wrapping paper will it take to cover all 6 sides of the new box? What is the volume of the new box?

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a. The number of rolls of wrapping paper it will take to cover the new box is 2¹/₄ rolls

b. The volume of the new rectangular box is 270 in³

Area of rectangular prism

Since the small box is a rectangular prism, its surface area, A = 2(lb + lh + bh) where

  • l = length,
  • b = width and
  • h = height.

Now, when the dimensions are tripled, its new area is A' = 2(LB + LH + BH) where

  • L = 3l,
  • B = 3b and
  • H = 3h.

So, A' = 2(LB + LH + BH)

= 2((3l)(3b) + (3l)(3h) + (3b)(3h))

= 2(3)(3)(lb + lh + bh)

= 18A

a. Number of rolls of wrapping paper

The number of rolls of wrapping paper it will take to cover the new box is 2¹/₄ rolls

Now, since we require 1/8 of a roll of wrapping paper to initially cover all 6 sides of the small box with dimensions, l,b and h. Then the number of rolls we require to cover the box when its dimensions are tripled are (since the area is proportional to the number of rolls)

N = 18 × 1/8

= 18/8

= 9/4

= 2¹/₄ rolls

The number of rolls of wrapping paper it will take to cover the new box is 2¹/₄ rolls

b. Volume of the new rectangular box

The volume of the new rectangular box is 270 in³

The volume of the rectangular box V = lbh where

  • l = length,
  • b = width and
  • h = height

When the dimensions are tripled, V' = LBH where

  • L = 3l,
  • B = 3b and
  • H = 3h.

So, V' = LBH

= (3l)(3b)(3h)

= 27lbh

= 27V

Now since the initial volume of the rectangular box, V = 10 in³

V' = 27V

V' = 27 × 10 in³

V' = 270 in³

So, the volume of the new rectangular box is 270 in³

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