In the first column, we entered the size of the square to be cut from
each corner [x]. In the second and third columns, we calculated the length
[25-2x] and width of the box [15-2x], respectively, and in the fourth column,
the volume of the box [x(25-2x)(15-2x)]. In the following table the square
increases in size by 0.5 in. each time.
The maximum volume is 513 cu. in., at x=3 in. To produce a box of 400
cu. in., x is either between 1.5 and 2 in. or between 4.5 and 5 in.
In order to calculate more accurately, we can use a smaller increment for
the size of the square, x. Thus below we started at x = 1.5 in. and increase
in steps of 0.05 in.
Now the maximum volume is 513.04 cu. in., at x=3.05 in. Also, to construct
a box of 400 cu. in., we need x either between 1.5 and 1.55 in. or between
4.75 and 4.8 in.
Again, we can use a smaller increment to improve the accuracy. Below we
started at 1.521 in. and increase in steps of 0.0003 in.
Now we see that the maximum volume is 513.05 cu. in., at x = 3.0461 in.,
and at x = 1.5249 in. or 4.7928 in. the volume of the box is approximately
400 cu. in.
Thus, we could continue to decrease the size of the increment to improve
the accurracy.