# algebra 2

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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?

• algebra 2 -

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

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