Answer:
height = 3.3 in
V = 694.848 in^3
Step-by-step explanation:
length of cardboard, l = 29 in
width of cardboard, w = 16 in
let the side of square is d.
so the length of box = 29 - 2y
width of box = 16 - 2y
height of box = y
Volume of box = length x width x height
V = (29 - 2y) x (16 - 2y) x y
V = 464 y - 90 y^2 + 4 y^3
dV/ dy = 464 - 180 y + 12y^2
For maxima and minima, dV/dy = 0
12y^2 - 180 y + 464 = 0
By solving
y = 3.3 in, 11.7 in
Now find double differentiation
d²V/dy² = 24 y - 180
Put, y = 3.3 in, it is - 100.8
Put, y = 11.7 in, it is + 100.8
So for maximum volume, y = 3.3 in
And the value of maximum volume
V = 464 (3.3) - 90(3.3 x 3.3) + 4 (3.3 x 3.3 x 3.3) = 1531.2 - 980.1 + 143.748
V = 694.848 in^3
