Take an 8.5 by 14-inch piece of paper and cut out four equal squares from the corners. Fold up the sides to create an open box. Find the dimensions of the box that has maximum volume. (Enter your answers as a comma-separated list. Round your answer to three decimal places.)

Answer 1

cut squares of length x

v = x(8.5-2x)^2 = 72.25x - 34x^2 + 4x^3
dv/dx = 72.25 -68x + 12x^2

max/min achieved at x = 17/12 or 17/4

Just knowing about the shape of cubics, it's clear that 17/12 is where the max volume occurs.

So, the box is 8.5-17/6 x 8.5-17/6 x 17/12 = 17/3 x 17/3 x 17/12