Problema Solution
A flat, rectangular piece of cardboard is 18” by 12”. A box (with no lid) is formed by cutting squares from each corner of the same size from each corner and folding up the sides. Using the side-length of cut-out squares as the independent variable, find an expression for the resulting volume of the box, give a graph over the full domain, and use the graph to estimate the side-length that gives maximum volume, accurate to the nearest quarter-inch.
Answer provided by our tutors
the side-length of cut-out squares: x
volume of the box: V
V = W*L*H = (18-2x)(12-2x)x = 4x^3 -60x^2 +216x
So V = 4x^3 -60x^2 +216x, the domain is 0V' = 12x^2 -120x +216
V'=0, so 12x^2 -120x +216=0
x = (10-sqrt(28))/2 = 2.35 in =2.25 in
When x = 2.25 in, V is max