Question

rectangular package to be sent by the u.s. postal service can have a maximum combined length and girth (perimeter of a cross section) of 180 inches. write the volume v of the package as a function of x.

Answer 1

The answer is the maximum volume of 11664 cubic inches is obtained for the dimensions $x=18$and $w=36$.

The first step is to define the variables used to measure the dimensions. Let's use the following system. Variable for square end of the package x Variable for the length of the package w The formula for the volume is given by

$V=x x w=x^{2} w$

and the formula for the dimension restrictions is the following.

Girth +Length =$4 x+w=108$

The next step is to solve for width w in terms of the other dimension x as follows.

w=108-4 w

This can be substituted into the volume formula. So, the formula for the volume is

$V=V(x)=x^{2}(108-4 x)=108 x^{2}-4 x^{3}$

Note that the dimensions are positive and the domain for the volume function is the interval [0,27].

Now we can determine the critical points for the function and then determine the dimensions that maximize the volume. The derivative of the volume function is

$\frac{d V}{d x}=216 x-12 x^{2}=12 x(18-x)$

The critical points as x=0 and x=18

For this problem, we can compute the volume for the critical points and the end points of the interval that is the domain of the volume function. This $x=0, w=108 \quad \mathrm{~V}(0)=0$

gives $x=18, w=36 \quad \mathrm{~V}(18)=11664$ absolute maximum As another test $x=27, w=0 \quad \mathrm{~V}(27)=0$

we could have used the first or second derivative tests to show that $x=18$is a relative maximum.

The maximum volume of 11664 cubic inches is obtained for the dimensions $x=18$and $w=36$.

What is volume?

• A volume is just the amount of space taken up by any three-dimensional solid. A cube, a cuboid, a cone, a cylinder, or a sphere are examples of solids.
• Volumes change depending on the form. In 3D geometry, we examined numerous three-dimensional forms and solids such as cubes, cuboids, cylinders, cones, and so on. We'll learn how to calculate volume for all of these forms.

To learn more volume about visit:

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