Answer:
12 inches
Explanation:
You want the width of an open-top box that is folded from a piece of cardboard with an area of 460 square inches. The box is 3 inches longer than wide, and squares of 4 inches are cut from the corners of the cardboard before it is folded to make the box.
Cardboard dimensions
The flap on either side of the bottom of width x is 4 inches, so the width of the cardboard is 4 + x + 4 = (x+8). The length is 3 inches more, so is (x+11).
The product of length and width is the area:
(x +8)(x +11) = 460 . . . . . . . . square inches
Solution
x² +19x +88 = 460
x² +19x -372 = 0
(x +31)(x -12) = 0 . . . . . . . factor
x = 12 . . . . . . . . . . . the positive value of x that makes a factor zero
The width of the box is 12 inches.
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Αdditional comment
The attached graph shows the solutions to (x+8)(x+11)-460 = 0. We prefer this form because finding the x-intercepts is usually done easily by a graphing calculator.
Another way to work this problem is to let z represent the average of the cardboard dimensions. Then the width is (z -1.5) and the length is (z+1.5) The product of these is the area: (z -1.5)(z +1.5) = 460. Using the "difference of squares" relation, we find this to be z² -2.25 = 460, the solution being z = √(462.25) = 21.5. Now, you know the cardboard width is 21.5 -1.5 = 20, and the box width is x = 20 -8 = 12.
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