Question

A parcel 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.

Answer 1

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

$V=xy^2$

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

$V=(120-4y)y^2=120y^2-4y^3$

Let us find the critical points of V

$\displaystyle\frac{\text{d}V}{\text{d}y}=240y-12y^2\\\\\displaystyle\frac{\text{d}V}{\text{d}y}=0\Rightarrow 240y=12y^2$

We can assume y>0, otherwise there is no parcel.

Dividing both sides by y:

$240=12y\Rightarrow y=\displaystyle\frac{240}{12}=20$

Since

$\displaystyle\frac{\text{d}^2V}{\text{d}y^2}=240-24y\Rightarrow \displaystyle\frac{\text{d}^2V}{\text{d}y^2}(20)=240-480<0$

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.