Answer:
x= 18
y= 18
z= 36
Step-by-step explanation:
The volume of the package = xyz
The girth (perimeter) of the package = 2(x+y)
Since the sum of the girth and length does not exceed 108, we have
z + 2(x+y) = 108
z = 108 - 2(x+y)
Put the value of z into V = xyz
V = xy(108 - 2(x+y)
V = xy (108 -2x - 2y)
V = 108xy - 2x^2y - 2xy^2
V= f(x,y)
Differentiate V with respect to x
dV/dx = 108y - 4xy - 2y^2 = 0
Differentiate V with respect to y
dV/dy = 108x - 4xy - 2x^2 = 0
Therefore we have
dV/dx = 108y - 4xy - 2y^2 = 0 and
dV/dy = 108x - 4xy - 2x^2 = 0
From the symmetry of both, x= y
Therefore;
108x - 4xy - 2x^2 = 0
108x - 4x(x) - 2x^2
108x - 4x^2 - 2x^2 = 0
108x - 6x^2 = 0
6(18x - x^2) = 0
18x - x^2 = 0
x(18 - x) = 0
Therefore, x= 0 or 18-x =0
x = 0 or x= 18
This means x=y=0 or x=y=18
V= f(x,y) has a critical point at (x, y) = (18,18)
Recall that z = 108 -2(x+y)
z = 108 - 2(18+18)
z = 108 - 2(36)
z = 108 - 72
z = 36
Therefore x=y=18 and z= 36
