 Calculus I optimization word problem - XP Math - Forums Sign Up FREE! | Sign In | Classroom Setup | Common Core Alignment  XP Math - Forums Calculus I optimization word problem 11-14-2009 #1 kermitt.the.frog   Join Date: Nov 2009 Posts: 1 Calculus I optimization word problem 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).  11-16-2009   #2
MAS1   Join Date: Dec 2008
Posts: 249 Max Volume Box

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  02-21-2010 #3 jmw106462      Join Date: Feb 2010 Posts: 388 Max Volume = 68.16  02-22-2010 #4 jmw106462      Join Date: Feb 2010 Posts: 388 Max Volume = 68.16 Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Posting Rules You may not post new threads You may not post replies You may not post attachments You may not edit your posts BB code is On Smilies are On [IMG] code is On HTML code is Off Forum Rules
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