The formula for the volume of the box and the value of x that will maximize the volume of the box are;
The volume of the box, is; V = 4·x³ - 38·x² + 88·x
The value of x that will maximize the volume of the box is; 3 1/6 inches
The steps used to analyze the volume of the box are as follows;
Let x represent the side length of the square;
The height of the open box = x
The length of the open box = 11 - 2·x
The width of the open box = 8 - 2·x
Volume of the box, V, is; x × (8 - 2·x) × (11 - 2·x) = 4·x³ - 38·x² + 88·x
The maximum volume can be found from the rate of change of the function for the volume
The maximum volume is the point where the rate of change of the volume function is zero, dV/dx = 0, therefore;
V = 4·x³ - 38·x² + 88·x
dV/dx = 12·x² - 76·x + 88
Therefore, at the x-value for the maximum volume, we get;
dV/dx = 0
12·x² - 76·x + 88 = 0
x = -(-76)/(2 × 12)
x = 19/6
19/6 = 3 1/6
The value of x that will maximize the volume of the box is x = 3 1/6 inches