a 16cm x 30cm sheet of rectangular cardboard is folded such that an open - top box is created. what is the…

Explanation:

Step1: Establish volume formula

The length of the box is $l = 30 - 2x$, the width is $w=16 - 2x$, and the height is $h = x$. The volume $V(x)=(30 - 2x)(16 - 2x)x=4x^{3}-92x^{2}+480x$.

Step2: Find the derivative

Using the power - rule $(x^n)'=nx^{n - 1}$, $V'(x)=12x^{2}-184x + 480$.

Step3: Set the derivative equal to zero

$12x^{2}-184x + 480 = 0$. Divide through by 4: $3x^{2}-46x + 120=0$. Using the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ for $ax^{2}+bx + c = 0$, here $a = 3$, $b=-46$, $c = 120$. So $x=\frac{46\pm\sqrt{(-46)^{2}-4\times3\times120}}{2\times3}=\frac{46\pm\sqrt{2116 - 1440}}{6}=\frac{46\pm\sqrt{676}}{6}=\frac{46\pm26}{6}$. We get $x=\frac{46 + 26}{6}=\frac{72}{6}=12$ or $x=\frac{46-26}{6}=\frac{20}{6}=\frac{10}{3}$. But $x = 12$ is not valid since $16-2x=16 - 24<0$. So we use $x=\frac{10}{3}$.

Step4: Calculate the volume

Substitute $x=\frac{10}{3}$ into the volume formula $V(\frac{10}{3})=(30-2\times\frac{10}{3})(16 - 2\times\frac{10}{3})\times\frac{10}{3}=(30-\frac{20}{3})(16-\frac{20}{3})\times\frac{10}{3}=(\frac{90 - 20}{3})(\frac{48 - 20}{3})\times\frac{10}{3}=(\frac{70}{3})(\frac{28}{3})\times\frac{10}{3}=\frac{70\times28\times10}{27}=\frac{19600}{27}\approx725$ $cm^{3}$.

Answer:

$725\ cm^{3}$