Question

a rectangular piece of cardboard measuring 14 inches by 27 inches is to be made into a box with an open top by cutting squares of equal size from each corner and golding up the sides . let x represent the length of a side of each square. for what value of x will the volume be a maximum? round to two decimal places.

Answer 1

v = x(14-2x)(27-2x)

dv/dx = 2(6x^2 - 82x + 189)

dv/dx=0 when x = (41 ± √547)/6 = 2.93, 10.73

knowing the shape of cubics, you should have no trouble "differentiating" the max from the min.

Answer 2

length = z = 27 -2x
width = y = 14 - 2x

v = (27-2x)(14-2x)x
v = ( 378 - 82 x + 4 x^2 ) x
v = 378 x -82 x^2 + 4 x^3

max when dv/dx = 0
dv/dx = 378 - 164 x + 12 x^2 = 0
6 x^2 - 82 x + 189 = 0

x = [ 82 +/- sqrt (6724-4536) ]/12

x = [82 +/- 46.8 ]/12

x = 10.7 too big, negative width
x = 2.93 inches