a candy box is made from a piece of cardboard that measures 16 by 9 inches. squares of equal size will be…

Explanation:

Step1: Define the variables

Let the side - length of the square cut from each corner be $x$ inches. The length of the box will be $l = 16 - 2x$ inches, the width will be $w=9 - 2x$ inches, and the height will be $h = x$ inches. The volume $V$ of the rectangular box is given by $V(x)=(16 - 2x)(9 - 2x)x$.

Step2: Expand the volume function

[ \begin{align*} V(x)&=(16 - 2x)(9 - 2x)x\ &=(144-32x - 18x+4x^{2})x\ &=(144 - 50x+4x^{2})x\ &=144x-50x^{2}+4x^{3} \end{align*} ]

Step3: Find the derivative of the volume function

Using the power rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, we have $V^\prime(x)=144-100x + 12x^{2}$.

Step4: Set the derivative equal to zero

We set $V^\prime(x)=0$, so $12x^{2}-100x + 144 = 0$. Divide through by 4 to simplify: $3x^{2}-25x + 36=0$.

Step5: Solve the quadratic equation

The quadratic formula for $ax^{2}+bx + c = 0$ is $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. Here, $a = 3$, $b=-25$, and $c = 36$. [ \begin{align*} x&=\frac{25\pm\sqrt{(-25)^{2}-4\times3\times36}}{2\times3}\ &=\frac{25\pm\sqrt{625-432}}{6}\ &=\frac{25\pm\sqrt{193}}{6}\ &=\frac{25\pm13.892}{6} \end{align*} ] We get two solutions: $x_1=\frac{25 + 13.892}{6}=\frac{38.892}{6}\approx6.48$ and $x_2=\frac{25 - 13.892}{6}=\frac{11.108}{6}\approx1.85$. But $x = 6.48$ is not valid since $9-2x=9 - 2\times6.48=9 - 12.96<0$.

Answer:

$1.85$