Quote:
Originally Posted by kermitt.the.frog
A box is to be made out of a 6 by 18 piece of cardboard. Squares of equal size will be cut out of each corner, and then the ends and sides will be folded up to form a box with an open top. Find the length (L) , width (W), and height (H) of the resulting box that maximizes the volume.
(Assume that W is < or = to L).

W = width
L = length
H = height
V = volume = W*L*H
S = length of the square cutout
W = 6  2S
L = 18  2S
H = S
V = (6  2S)(18  2S)(S)
V = (6  2S)(18S  2S^2)
V = 108S  36S^2  12S^2 + 4S^3
V = 4S^3  48S^2 + 108S
To maximize the volume, take the derivative and set it equal to 0.
der(V) = 12S^2  96S + 108
0 = 12S^2  96S + 108
0 = 12(S^2  8S + 9)
0 = S^2  8S + 9
Use the quadratic formula to solve for S.
S = (8 + sqrt((8)^2  4(1)(9)))/2(1)
S = (8 + sqrt(28))/2
S = 4 + sqrt(7)
S = 4 + sqrt(7) gives a value greater than 6 which cannot be used since the width is only 6. Therefore:
S = 4  sqrt(7) which is about 1.35425
Plug back in to find W, L, H, and V.
W = 6  2(1.35425) = 3.2915
L = 18  2(1.35425) = 15.2915
H = 1.35425
Max Volume = 68.16