SOLUTION: Suppose that an open box is to be made from a square sheet of cardboard by cutting out 3x3 squares from each corner and then folding along the dotted lines. If The box is to have a

Question 158774: Suppose that an open box is to be made from a square sheet of cardboard by cutting out 3x3 squares from each corner and then folding along the dotted lines. If The box is to have a volume of 300 cubic inches, find the origional dimensions of the sheet of cardboard.
This is from a take home exam and I am stumped !
Answer by gonzo(654) (Show Source): You can put this solution on YOUR website!
if the box is going to have a volume of 300 cubic inches, then length * width * height must = 300.
looking at a sheet of paper and visualizing 3 inch squares cut out on each end, it looks to me like the height of the box will be 3 inches because the section left in the middle will form the bottom of the box and the sections left hanging out of each side of the middle will form the height of the box as they are folded up.
so the height of the box will be 3 inches which leaves the dimensions of the bottom of the box.
if the volume of the box is 300 cubic inches, then 300 / 3 inch height leaves a bottom that is 100 square inches.
that 100 square inches forms the area of the bottom of the box which is a product of length of the box times width of the box.
since there are no restrictions on the length in relationship to the width, i would suspect that the length and width could be anything as long as their product equals 100 (length * width).
it could be 10 * 10.
it could be 5 * 20.
it could be 100 * 1.
i will assume a bottom of the box as being 10x10 = 100 square inches. this makes further calculations simpler and satisfied the requirements of the problem.
length of the box is therefore 10 inches and width of the box is therefore 10 inches.
volume is 3*10*10 = 300 which satisfies the requirements for the volume of the box.
getting back to the original sheet of paper, if what is left of the width of the sheet of paper must equal 10 inches and what is left of the length of the sheet of paper must also equal 10 inches, then adding 3 inches to each end of it results in an original length of 16 inches in length and 16 inches in width.
prove this by drawing a square 16 inches in length and 16 inches in width.
if you don't have a piece of paper that big then draw it to scale, i.e. make a 1/2 inch be equivalent to an inch.
then 8 inches would be equivalent to 16 inches and 1.5 inches would be equivalent to 3 inches.
draw a 16 inch square.
draw a 3 inch square at each corner.
visualize cutting out the corners and folding up each resulting side that is hanging out.
the resulting box will equal 10 inches in length and 10 inches in width and 3 inches in height.
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