Width = W
Length = 2W
Height = h
Volume = 2300 cm^{3} = L*W*h = 2W*W*h = 2W^{2}h
So h = 2300/2W^{2} = 1150/W^{2}
Area = 2*(W*h) + 2*(2W*h) + W*2W = 6900/W + 2W^{2} [Substituted 1150/W^{2} for h, L = 2W]
To find the minimum Area, take the derivative of the AREA wrt W, set it to zero, and solve for W:
d(Area)/dW = 6900W^{2} + 4W
0 = 6900W^{2} + 4W
6900/4 = W^{3}
1725 = W^{3}
12 ≅ W [11.993 rounded to 12]
L = 2W = 24 [23.986 rounded to 24]
h = 1150/W^{2} = 8 [7.995 rounded to 8]
CHECK:
Volume = 12*24*8 = 2304 [Within rounding error]
6/19/2014

Philip P.