Aparcel delivery service will deliver a package only if the length plus the girth (distance around, i. e., perimeter, taken perpendicular to the length) does not exceed 120 inches. find the maximum volume of a rectangular box with square ends that satisfies the delivery company's requirements.

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.