Expert: Ahmed Salami - 10/10/2008

Question
An open box is to be made from a twelve-inch by twelve-inch square piece of material by cutting equal squares from the corners and turning up the sides (see figure below, where y = 12 − 2x). Find the volume of the largest box that can be made.

Answer
Hi Gracie,
I cant see the diagram but i picture the situation. You end up with a box with a height x and a square base of length y = 12 - 2x. The volume of the box can therefore be expressed as
V = xy^2
 = x(12 - 2x)^2
 = x(144 - 48x + 4x^2)
 = 144x - 48x^2 + 4x^3
To find the largest possible volume we have to find the maximum value of this function by optimizing. We achieve this by finding the differential with respect to x and equating to zero.
V = 144x - 48x^2 + 4x^3
dV/dx = 144 - 96x + 12x^2
equating to zero
12x^2 - 96x + 144 = 0
dividing through by 12
x^2 - 8x + 12 = 0
(x - 2)(x - 6) = 0
x = 2 or 6
Now we have two values of x which, in this case, corresponds to a maximum and a minimum value for the volume. To identify each one of them we simply investigate them at the second differential. A negative second differential value means the point is a maximum and vice versa.
The second differential,
d^2V/dx^2 = -96 + 24x
at x = 2
d^2V/dx^2 = -96 + 48
         = -48
which confirms that the point x = 2 gives us a maximum value for V.
at x = 6
d^2V/dx^2 = -96 + 144
         = 48
which confirms that the point x = 6 gives us a minimum value for V.
Therefore, we get the largest volume for the box when we take x to be 2 inches. And by implication,
y = 12 - 2x
 = 12 - 4 = 8 inches
So, the largest volume is
V(max) = xy^2
      = 2(8^2)
      = 2(64)
      = 128 cubic inches
OR
V(max) = 144x - 48x^2 + 4x^3
      = 144(2) - 48(4) + 4(8)
      = 288 - 192 + 32
      = 128 cubic inches
I hope its alright. You can always do your own cross-check and i'm open for further explanation.
Regards.