a box (with no top) will be made by cutting squares of equal size out of the corners of a 31 inch by 48 inch…

Explanation:

Step1: Define the variables

Let $x$ be the side - length of the square cut from each corner. The length of the box will be $l = 48 - 2x$, the width will be $w=31 - 2x$, and the height will be $h = x$. The volume $V$ of the box is given by the function $V(x)=(48 - 2x)(31 - 2x)x$.

Step2: Expand the function

[ \begin{align*} V(x)&=(48 - 2x)(31 - 2x)x\ &=(1488-96x - 62x+4x^{2})x\ &=(1488 - 158x+4x^{2})x\ &=1488x-158x^{2}+4x^{3} \end{align*} ]

Step3: Find the derivative

Using the power rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, we have $V^\prime(x)=1488-316x + 12x^{2}$.

Step4: Set the derivative equal to zero

$12x^{2}-316x + 1488 = 0$. Divide through by 4: $3x^{2}-79x + 372=0$. Using the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ for the quadratic equation $ax^{2}+bx + c = 0$, here $a = 3$, $b=-79$, $c = 372$. [ \begin{align*} x&=\frac{79\pm\sqrt{(-79)^{2}-4\times3\times372}}{2\times3}\ &=\frac{79\pm\sqrt{6241-4464}}{6}\ &=\frac{79\pm\sqrt{1777}}{6}\ &=\frac{79\pm42.15}{6} \end{align*} ] We get $x_1=\frac{79 + 42.15}{6}\approx20.2$ and $x_2=\frac{79 - 42.15}{6}\approx6.14$. But $x = 20.2$ is not valid since $31-2x=31 - 2\times20.2=-9.4\lt0$. So we consider $x\approx6.14$.

Step5: Calculate the volume

Substitute $x\approx6.14$ into the volume formula $V(x)=(48 - 2x)(31 - 2x)x$. $l=48-2\times6.14 = 48 - 12.28=35.72$, $w=31 - 2\times6.14=31 - 12.28 = 18.72$, $h = 6.14$. $V=(35.72)\times(18.72)\times(6.14)\approx4088$

Answer:

4088