A manufacturer is making cardboard boxes by cutting out for equal squares from the corners of the rectangular piece of cardboard and then folding the remaining part into a box.

Question

The length of the cardboard piece is 1 in. longer than its width. The manufacturer can cut out either 3 x 3 in. squares or 4 x 4 in. squares. Find the dimensions of the cardboard for which the volume of the boxes produce by both methods will be the same.

Answer 1

L=1+W

cut corners of 3x3

means box sides of L-6 = W-5 and W-6

Volume = box length x box width x box height

Volume = (W-5)(W-6)H = (W^2 -11W +30)3

= 3W^2 -33W+90

cut corners 4x4 means H=4, box sides = W-8 and L-8 = W-7

Volume = (W-7)(W-8)4

= 4(W^2 -15W +56)

= 4W^2 -60W +224

both volumes are equal when

3W^2 -33W +90 = 4W^2 -60W +224

W^2 -27W + 134 =0

use the quadratic formula

W = 27/2 +/-(1/2)sqr(27^2 -4(1)(134))

W = 13.5 +/-.5sqr193 = about 13.5+/-13.892/2

W = about 13.5 +/-6.946

W = about 20.446 or 6.554 inches Wide

L = about 21.446 or 7.554 inches Long

for cutting corners 3x3 or 4x4 and having the volume equal

L= about 21.4 or 7.6

W = about 20.4 or 6.6