Question

Aimee is shipping a care package to her friend in college, and she has a piece of cardboard with which to create a box. A rectangular cardboard sheet measuring 20 inches by 40 inches is to be used by cutting out square corners of measure x inches.

Answer 1

The volume of the box will be given by:

`V=x(20-2x)(40-2x)`

And the according graph is:

To find the value of x that provide the maximum value of the box, we could find the deritative of the volume function:

`V=x(20-2x)(40-2x)`

First, let's rewrite the function as:

`V(x)=4x^3-120x^2+800x`

The deritative of this function is:

`(dV)/(dx)=12x^2-240x+800`

Now, we're going to equal this equation to zero and then solve for x:

`\begin{gathered} (dV)/(dx)=0 \\ \\ 12x^2-240x+800=0 \\ Solving\colon \\ x=\begin{cases}\frac{30-10\sqrt[]{3}}{3} \\ \frac{30+10\sqrt[]{3}}{3}\end{cases} \end{gathered}`

We obtained two solutions which are the critical points of V. The first solution is the point that shows a maximum in the graph of V(x), so, the value of x that provide the maximum value of the graph is:

`x=\frac{30-10\sqrt[]{3}}{3}`