Question

A rectangular package sent by a postal service can have a maximum combined length and girth (perimeter of a cross sectio) of 108 inches. Find the dimensions of the package of maximum volume that can be sent, if the cross sectional region must be square.

Answer 1

Answer:

The maximum volume of the package is obtained with a cross section of side 18 inches and a length of 36 inches.

Step-by-step explanation:

This is a optimization with restrictions problem.

The restriction is that the perimeter of the square cross section plus the length is equal to 108 inches (as we will maximize the volume, we wil use the maximum of length and cross section perimeter).

This restriction can be expressed as:

$4x+L=108$

being x: the side of the square of the cross section and L: length of the package.

The volume, that we want to maximize, is:

$V=x^2L$

If we express L in function of x using the restriction equation, we get:

$4x+L=108\\\\L=108-4x$

We replace L in the volume formula and we get

$V=x^2L=x^2*(108-4x)=-4x^3+108x^2$

To maximize the volume we derive and equal to 0

$\dfrac{dV}{dx}=-4*3x^2+108*2x=0\\\\\\-12x^2+216x=0\\\\-12x+216=0\\\\12x=216\\\\x=216/12=18$

We can replace x to calculate L:

$L=108-4x=108-4*18=108-72=36$

The maximum volume of the package is obtained with a cross section of side 18 inches and a length of 36 inches.

Answer 2

The dimensions that maximize the volume of the package are 18 inches by 36 inches

Represent the length of the square cross-section with x, and the length of the package with y.

So, we have:

$\mathbf{y + 4x = 108}$ --- perimeter

Make y the subject

$\mathbf{y = 108 - 4x}$

The volume is calculated as:

$\mathbf{V = x^2y}$

Substitute $\mathbf{y = 108 - 4x}$ in $\mathbf{V = x^2y}$

$\mathbf{V =x^2(108 - 4x)}$

Expand

$\mathbf{V =108x^2 - 4x^3}$

Differentiate

$\mathbf{V' =216x - 12x^2}$

Set to 0

$\mathbf{216x - 12x^2 = 0}$

Rewrite as:

$\mathbf{12x^2 = 216x}$

Divide both sides by 12x

$\mathbf{x = 18}$

Recall that:

$\mathbf{y = 108 - 4x}$

So, we have:

$\mathbf{y = 108 - 4(18)}$

$\mathbf{y = 36}$

Hence, the dimensions that maximize the volume of the package are 18 inches by 36 inches

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