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Maria has a piece of cardboard that is twice as long as it is wide. If she cuts out 2-in squares from each corner and bends up the side to form a box with no top, she will have a box with a volume of 140cubic inches. Find the dimensions of the piece of cardboard.

Sagot :

Uncut piece of cardboard:
Width: x inches
Length: 2x inches

Piece of cardboard with four cut-out corners:
Subtract 2 × 2 inches from length and width to compute for the volume of the box.
Width: x - 4 inches
Length: 2x - 4 inches
Height: 2 inches
Volume: 140 cubic inches

Equation:
Length × Width × Height = Volume
(2x-4) (x - 4) (2) = 140
(2x² - 8x - 4x + 16) (2) = 140
(2x² - 12x + 16) (2) = 140
4x² - 24x + 32 = 140

Quadratic equation, ax² + bx + c = 0
4x² - 24x + 32 - 140 = 0
4x² - 24 - 108 = 0

Solve by factoring:
4 (x² - 6x - 27) = 0
4 (x - 9) (x + 3) = 0

x - 9 = 0                   x + 3 = 0
x = 9                        x = -3

Choose the positive root x = 9.

The dimensions of the piece of cardboard (uncut) are:
Width: x = 9 inches
Length: 2x = 2(9) = 18 inches

The dimensions are 9 inches and 18 inches.

Check:
(18-4) (9-4) (2) = 140
(14) (5) (2) = 140
140 = 140
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