Question 593744: The area of the rectangular piece of cardboard shown below is 216 square inches. The cardboard is used to make an open box by cutting a 2-inch square from each corner and turning up the sides. If the box is to have a volume of 224 cubic inches, find the length and width of the cardboard that must be used.
Answer by ankor@dixie-net.com(22740)   (Show Source):
You can put this solution on YOUR website! The area of the rectangular piece of cardboard shown below is 216 square inches.
The cardboard is used to make an open box by cutting a 2-inch square from each corner and turning up the sides.
If the box is to have a volume of 224 cubic inches, find the length and width of the cardboard that must be used.
:
Let L = the length of the box
then
(L+4) = the length of the piece cardboard (because of the 2" squares)
and
Let W = the width of the box
then
(W+4) = the width of the cardboard
:
From the information given we know the height of the box will be 2", therefore
the area of the bottom of the box will be: 224/2 = 112 sq/in
L*W = 112
and
(L+4)*(W+4) = 216, the area of the cardboard
FOIL
LW + 4L + 4W + 16 - 216 = 0
LW + 4L + 4W - 200 = 0
Replace LW with 112
112 + 4L + 4W - 200 = 0
4L + 4W - 200 + 112 = 0
4L + 4W - 88 = 0
simplify, divide by 4
L + W - 22 = 0
L + W = 22
L = (22-W)
Replace L with (22-W) in the area of the base equation
(22-W)*W = 112
-W^2 + 22W = 112
Arrange as a quadratic equation on the right
0 = W^2 - 22W + 112
Factors to
(W-8)(W-14) = 0
Two solutions
W = 8, then L = 14
W = 14, then L = 8
:
The box is 14 by 8 by 2, check volume: 14*8*2 = 224 cu/in
;
We add 4" to the length and width of the cardboard: 18" by 12".
Check the area: 18*12 = 216 sq/in
:
Could you follow all this? Did I make it understandable to you? C
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