Answer:
16,000 cubic inches.
Step-by-step explanation:
Let x, y, z be the length, width and height of the parcel.
Since the parcel is a rectangular box with square ends,
y = z and the girth is 4y.
The volume is
The maximum volume must occur when the length plus the girth equals 120 inches, that is
x + 4y = 120
We want then to find the maximum of the volume.
As
x = 120 - 4y
we have
Let us find the critical points of V
We can assume y>0, otherwise there is no parcel.
Dividing both sides by y:
Since
and
y=20
is a maximum. For this value of y:
x = 120 - 4(20) = 40
We conclude that the rectangular box with maximum volume that satisfies the delivery company's requirements has the following measures:
Length = 40
Width = height = 20 inches
Volume = 40*20*20 = 16,000 cubic inches.