Question

you can make an open box from a piece of flat cardboard. First cut congruent squares from the four corners of the cardboard. Then fold and tape the sides. let x equal the side of each congruent squares as x increases so does the depth of the box the useable area of the cardboard decreases as x increases, and so do the length and width of the box. what happens to the volume of the box? does it increase or deacreas as x increases? would the answer both suprise you? what size square should you cut from the corners to maximize the volume of your box?What are the dimensions of the box in centimeters?

Answer 1

length of bottom = L-2x
width of bottom = W -2x

v = x (L-2x)(W-2x)
= x(LW-2Wx-2Lx +4x^2)
= LW x -2Wx^2 -2Lx^2 +4x^3

This is a cubic polynomial and will have maxima and minima. Just looking at it the volume would be huge as x got huge because of the x^3. However x can not be bigger than W/2 in practice so it is not that simple.
Using calculus to find max or min:
dv/dx = LW -4Wx-4Lx + 12 x^2
that is 0 at a max or min
12 x^2 -4(L+W)x + LW = 0
for a given L and W, solve that quadratic for x of min or max
for example for a square sheet of 10 cm on a side

12 x^2 -4(20) + 100 = 0
3 x^2 -20 + 25 = 0
x = [20 +/- sqrt (400 -300) ]/6
x = [20 +/-10]/6
x = 30/6 or 10/6
30/6 is 5 which is half the width so zero volume
so 10/6 is what we have, 1 2/3 deep