Question 1174860
You have a flat sheet of cardboard that is 10 inches long and 8 inches wide.
 You need to make a box that has the maximum volume.
 So you decide to cut out square corners of the sheet of cardboard, and fold the cardboard to make a box.
 Let x represent the side of each little square you cut out.
 Write a function for the volume of the box and use your graphing calculator to graph it.
What is the maximum volume of your box, and what value of x gives you that maximum area?
 Be sure to justify your answer.
:
the dimensions of the box will be
(10-2x) by (8-2x) by x, which is the height
f(x) = Volume
V = (10-2x)*(8-2x)*x
FOIL
v = (80-20x-16x+4x^2)*x
V = (80-36x+ 4x^2)*x
multiply by x, arrange in the standard order
V = 4x^3 - 36x^2 = 80x
Graph this equation, volume is on the vertical axis, x is the horizontal
{{{ graph( 300, 200, -2, 5, -10, 70, 4x^3-36x^2+80x, 52.5) }}}
calc shows max volume is when x = 1.47 inches and 
then the max vol = 52.5 cu/in (green line)
:
You can confirms this by replacing 1.47 for x in the original equation, and finding the volume