SOLUTION: Rick has a rectangular piece of cardboard that is twice as long as it is wide. From each of the 4 corners, Rick cuts a 2in square and turns up the flaps to form an uncovered box.

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Question 280710: Rick has a rectangular piece of cardboard that is twice as long as it is wide. From each of the 4 corners, Rick cuts a 2in square and turns up the flaps to form an uncovered box. If the volume of the box is 896in^3, what were the dimensions of the original piece of cardboard?
Found 2 solutions by checkley77, oberobic:
Answer by checkley77(12844) About Me  (Show Source):
You can put this solution on YOUR website!
X*2X=2X^2 is the original dimensions of the cardboard.
W*L*H=VOLUME
W=(x-2*2)=x-4 in.
L=(2x-2*2)=2x-4 in.
H=2 in.
(x-4)(2x-4)*2=896
2(2x^2-8x-4x+16)=896 divide by 2
2x^2-12x+16=448
2x^2-12x+16-448=0
2x^2-12x-432=0
2(x^2-6x-216)=0
2(x-18)(x+12)=0
x-18=0
x=18
18-4=14 one side after corner cuts.
2*18=36-4=32 the other side after corner cuts.
Proof:
14*32*2=896
896=896


Answer by oberobic(2304) About Me  (Show Source):
You can put this solution on YOUR website!
l = 2w :: length is twice the width
.
New box dimensions:
.
l-4 = length - 2 corners cut away at 2 inches each
w - 4 = w - 2 corners cut away at 2 inches each
(l-4)(w-4)*2 = volume = 896 cubic inches
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substituting l = 2w
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(2w-4)(w-4)*2 = 896
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divide both sides by 2
(2w-4)(w-4) = 896/2 = 448
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collect and simplify
2w^2 -8w -4w +16 - 448 = 0
2w^2 -12w -432 = 0
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divide by 2 to further simplify
w^2 -6w - 216 = 0
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Thinking about factoring...
216 = 2*108 = 2*2*54= 2*2*2*27 = 2*2*2*3*9
2*2*3 = 12
2*9 = 18
12 & 18 are 6 apart,
So the factoring is:
(w-18)(w+12) = 0
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Roots are w = 18 and w = -12.
But negative width is nonsense, so we think w = 18.
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Substituting in l = 2w
l = 2*18 = 36
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Thus we think the original dimensions of the cardboard are length = 36 and width = 18.
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Checking our work, what is the volume of the box?
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Recall the volume of the box is based on different dimensions:
l-4 = 36-4 = 32
w-4 = 18-4 = 14
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is the volume of a box 32 by 14 by 2 = 896?
Yes it is!
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Answer:
The original dimensions of the box are length = 36 and width = 18.
.
Done