Optimization problem?

Question

A rectangular box with open top is to be formed from a rectangular piece of cardboard which is 192 inches by 140 inches by cutting a square of size x inches from each corner, and folding up the sides. What is the largest possible volume of such a box?

To solve this problem we need to maximize the following function of x:
f(x)=
over the interval from____ to____
We find that f(x) has a stationary point at x=
To verify that f(x) has a maximum at this stationary point we compute the second derivative f''(x) and find that its value at the stationary point is ______, a negative number.
Thus the maximum volume of the box is:

Diagram of the problem: http://i144.photobucket.com/albums/r162/patel748/5.png

Answer 1

Maximize your volume f(x) = (192 - 2x)(140 - 2x)x = 26880x - 664 x^2 + 4x^3 for 0<=2x<=140 or 0<=x<=70.
df/dx = 26880 - 1328x + 12x^2 = 0 which gives x1 = 10.63 and x2 = 210.7. Since x2 > 70 x1 is a solution.
Second derivative at this point d2f/dx2 = -1328 + 24*x1 = 1072.88 < 0 , so this is the maximum.
Therefore the maximum volume = f(x1) = 215509.0867.