Respuesta :
An open box is to be made out of a 6-inch by 14-inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. Find the dimensions of the resulting box that has the largest volume.
Dimensions of the bottom of the box: L x W
Height of the box:
SOLUTION
Diagram
Cardboard with x2 area cutouts...
··················14 - 2x
···x··¦¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¦···x··
¦——¦·······························¦——¦
¦·············································¦
¦·············································¦ 6 - 2x
¦·············································¦
¦——¦·······························¦——¦
···x··¦___________________¦··x···
Volume = l·w·h
After cutouts:
l = 14 - 2x
w = 6 - 2x
h = x
V = (14 - 2x)(6 - 2x)(x) in3
V = 4x3 - 40x2 + 84x in3
To maximize the volume, take the derivative of V with respect to x, set it equal to zero, and solve for x:
dV/dx = 0 = 12x2 - 80x + 24
x2 - (20/3)x + 7 = 0
x = (10 ± √37)/3 in
Since (6 - 2x) yields a negative number for x = (10 + √37)/3 we use only x = (10 - √37)/3
Thus, the dimensions are:
l = 14 - 2x = 11.39 in
w = 6 - 2x = 3.39 in
h = x = 1.31 in
Base = l·w = 38.59 in2
Volume = 50.39 in3
