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.
:
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