Expert: Scotto - 10/19/2009

Question
An open box of maximum volume is to be made from a standard size poster board, by cutting equal squares from the corners and turning up the sides.
1. Purchase a standard size poster board(22"x28")
2. Make a 5 sided box by cutting out squares in the corners
3. The object is to make the box with maximum volume
4. Write the volume as a function of x.
5. Find the maximum volume
* I cut my box's in the corners by 4inches

Answer
The amount cut out of each corner is x•x.  

There are two corners in each side, so take of 2x from the length of each side.

This make the sides 22-2x and 28 -2x.

That makes the volume (22-2x)(28-2x)x.

Multiplying that out gives the volume V(x) = x³ - 50x² + 616x.

Using this, V'(x) = 3x² - 100x + 616.

Using the quadratic equation x = (-b±√(b²-4ac))/(2a) with a=3, b=-100, and c=616 gives
x = (100±√(10,000-4*3*616)/6 = (100±√2,608)/6.
Now an estimate of √2608 is 51, since 51²=2,601.
This means we should get value close to 8 and 25.
The real values come out as 25.17809689 and 8.155236443.

Checking these out with the function gives
   x                 V(x)
8.154236443   2240.61905
8.155236443   2240.619075
8.156236443   2240.61905
  
25.17709689   -225.804235
25.17809689   -225.8042605
25.17909689   -225.804235

As can be seen, the firt value gives us the maximum.

The second value is over half the length of the short side,
so it will result in a negative length, giving a negative volume.
This is the least it could be - mathematically, that is.

Checking V"(x) = 6x - 100, the value of V"(8) = -52 and the value of V"(25) = 50.
Since the second derivative at the 1st point is negative, this means the 1st is a maximum.
Since the second derivative at the 2nd point is positive, this means the 2nd is a minimum.