SOLUTION: An open box is to be constructed from a 12 x 12 inch piece of board by cutting away squares of equal size from the four corners and folding up the sides. Determine the size of the

Algebra ->  Volume -> SOLUTION: An open box is to be constructed from a 12 x 12 inch piece of board by cutting away squares of equal size from the four corners and folding up the sides. Determine the size of the       Log On


   

Question 1200116: An open box is to be constructed from a 12 x 12 inch piece of board by cutting away squares of equal size from the four corners and folding up the sides. Determine the size of the cut-out that maximizes the volume of the box?
A. 350
B. 274
C. 231
D. 128
Found 2 solutions by Fombitz, ikleyn:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Starting with the 12 x 12 box, let's call the length of the cutout, X.
So then the width of the side becomes L=12-2X since there are two cutouts. Since it's a square cutout off of a square, the height and width are the same.

When you fold up the open top box, you get the dimensions you need to find the volume of the box.

So then the volume of the box becomes,
V=%2812-2X%29%2812-2X%29X
V=%284X%5E2-48X%2B144%29X
V=4X%5E3-48X%5E2%2B144X
Since you have the volume as a function of one variable, you can take the derivative and set it equal to zero to find the extrema.
dV%2FdX=12X%5E2-96X%2B144=0
X%5E2-8X%2B12=0
%28X-6%29%28X-2%29=0
So there are two solutions, X=6 and X=2.
You can use the second derivative test to find which value gives you the maximum or the minimum.
As an alternative, you can also just plug the value into the volume equation.
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.
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In this case, neither value matches any of your choices. So I presume the choices are the maximum volume and not the size of the cutout. You can verify.

Answer by ikleyn(52771) About Me  (Show Source):
You can put this solution on YOUR website!
.
An open box is to be constructed from a 12 x 12 inch piece of board by cutting away squares of equal size
from the four corners and folding up the sides. Determine the size of the cut-out that maximizes the volume of the box?
A. 350
B. 274
C. 231
D. 128
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Tutor  Fombitz just noticed that the answers list does not match to the question.


I want to confirm it for new generations of students,  who may read this post in the future:

the problem is posed  INCORRECTLY,  the answers list is not consistent with the question.


I address my note to the future generation of students,  but not to the persons who created/submitted this post,

because,  in my view,  it is useless to explain anything to those who compose such gibberish:
they either do not understand what they post,  or do not read it,  at all.


                                        Or  BOTH.