SOLUTION: By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made (see the accompanying figure). If

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Question 1205671: By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made (see the accompanying figure). If the cardboard is l = 16 in. long and w = 9 in. wide, find the dimensions of the resulting box if it is to have a total surface area of 128 in.2
Found 2 solutions by Theo, josgarithmetic:
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
the total surface of the rectangular piece of cardboard, before cutting, is equal to 16 * 9 = 144 square inches.

if you remove a square of x inches on a side from each corner of the box, the remaining surface area will be 16 * 9 - 4 * x^2 = 144 - 4 * x^2

if the remaining surface area is 128 square inches, then your formula becomes 144 - 4 * x^2 = 128.

add 4 * x^2 to both sides and subtract 128 from both sides of the equation to get 144 - 128 = 4 * x^2 which simplifies to 16 = 4 * x^2 which simplifies further to 4 = x^2.

solve for x to get x = 2.

the length will be 16 minus 2 * 2 = 12
the width will be 9 minus 2 * 2 = 5
the height will be 2.

the area of the bottom of the box will be 12 * 5 = 60
the area of the 2 vertical sides along the length will be 12 * 2 = 24 each for a total area of 48.
the area of the 2 vertical sides along the width will be 5 * 2 = 10 each for a total area of 20.

the total surface area of the box will be 60 + 48 + 20 = 128 square inches.


Answer by josgarithmetic(39615) About Me  (Show Source):
You can put this solution on YOUR website!
Original Dimensions cardboard 16 in. long, 9 in. wide
Want box to have surface area 128 square inches
cut squares from the corners of x inches each.

EACH area surface pieces
%2816-2x%29%289-2x%29, x%2816-2x%29 two of these, x%289-2x%29 two of these;

Account for the total area (assuming only one side of surface; not both sides.)
%2816-2x%29%289-2x%29%2B2x%2816-2x%29%2B2x%289-2x%29=128
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Try to simplify what you can, first.
2%288-x%29%289-2x%29%2B2%28x%2816-2x%29%2Bx%289-2x%29%29=2%2A64
%288-x%29%289-2x%29%2B%2816x-2x%5E2%2B9x-2x%5E2%29=64
72-9x-16x%2B2x%5E2%2B25x-4x%5E2=64
72-2x%5E2=64
36-x%5E2=32
x%5E2=4
highlight%28x=2%29-------------Use this to find the dimensions of the box.