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# From a 12-cm by a 12-cm piece of cardboard, square corners are cut out so that the sides can be folded up to make a box.a) Express the volume of the box as a function of the side x, in centimeters, of a cut-out square.b) Find the domain of the function.c) Graph the function with a graphing calculator.d) What dimensions yield the maximum volume?

Solution

Verified

a)



The volume of the box is:

$V=lwh$

Since $l=12-2x$, $w=12-2x$, and $h=x$, the volume is:

$V(x)=(12-2x)(12-2x)(x)$

$V(x)=(144-48x+4x^2) (x)$

$V(x)= 144x-48x^2+4x^3$

or

$\color{#c34632}V(x)= 4x^3-48x^2+144x$



b)



Each dimension of the box must be greater than 0 so:

$12-2x>0\to x<6$

$12-2x>0\to x<6$

$x>0$

So, the domain is:

$\color{#c34632}0



c)



Graph the function from (a) using the domain from part (b): d)



Using the maximum feature, the maximum point is $(2,128)$. So, the volume is maximum at $x=2$ and the dimensions are:

\begin{aligned} \text{length}&=12-2x=12-2(2)=\color{#c34632}8\text{ cm}\\ \text{width}&=12-2x=12-2(2)=\color{#c34632}8\text{ cm}\\ \text{height}&=x=\color{#c34632}2\text{ cm} \end{aligned}

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