Precalculus
posted by John on .
You are given a 12"x18" piece of construction paper. You are to cut a square out of each corner in order to create a 3dimensional opentop box. What size squares would you need to cut in order to maximize the volume of the box?

let the side of the square to be cut out be x inches
resulting box is (122x) by (182x) by x
V = x(122x)(182x)= x(216  60x + 4x^2)
= 4x^3  60x^2 + 216x
dV/dx = 12x^2  120x + 216
= 0 for a max of V
x^2  10x + 18=0
solve for x 
length = (18  2x)
width = (12  2x)
V = x (182x)(122x)
= x(4)(9x)(6x)
= 4 x (5415x+x^2)
= 4 (54 x 15x^2 + x^3)
dV/dx = 0 for max
0 = 54  30 x + 3 x^2
0 = 18  10 x + x^2
x = [ 10 +/ sqrt (10072) ]/2
[ 10 +/ 2 sqrt 7 ]/2
= 5 +/ sqrt 7
5+sqrt 7 is no good, more than half the width
so
5sqrt 7 = 2.35 
Looks like calculus not precalculus to me by the way. I do not see how to do it without taking the derivative.

what does sqrt mean?