Question

a box is to be made out of a rectangular piece of cardboard that is 12 inches wide and 16 inches long. squares x inches on a side are cut out of the corners and the sides are bent upwards.

the length of the box is $16-2x$ in. (type an expression using x as the variable.)
the width of the box is $12-2x$ in. (type an expression using x as the variable.)
the height of the box is $x$ in. (type an expression using x as the variable.)
b. write a simplified expression for the volume of the box.
the volume of the box is $square$ $in^3$.
(type an expression using x as the variable.)

Explanation:

Step1: Recall volume formula for box

Volume $V = \text{length} \times \text{width} \times \text{height}$

Step2: Substitute given expressions

$V = (16-2x)(12-2x)(x)$

Step3: Multiply first two binomials

First, expand $(16-2x)(12-2x)$:

$$\begin{align*} (16-2x)(12-2x)&=16\times12 -16\times2x -2x\times12 + (-2x)(-2x)\\ &=192 -32x -24x +4x^2\\ &=4x^2 -56x +192 \end{align*}$$

Step4: Multiply by height $x$

$V = x(4x^2 -56x +192)$
$V = 4x^3 -56x^2 +192x$

Answer:

$4x^3 -56x^2 +192x$