Question

bisa has a piece of construction paper that she wants to use to make an open - rectangular prism. she will cut a square with side length x from each corner of the paper, so the length and width is decreased by 2x as shown in the diagram. answer parts a to d.

c. what do the x - intercepts of the graph mean in this context?
the intercepts 0 and 4 represent the side lengths of the cut squares that will result in a box with zero volume. the intercept 5.5 is not meaningful because it is not possible to cut this length from each corner of an 8 - inch side.

d. if bisa wants to maximize the volume of the box, what is the side length of the squares that should be cut from each corner of the piece of construction paper? explain.
about
(type a whole number.)

Explanation:

Step1: Find the volume formula

The length of the box after cutting is $l = 11 - 2x$, the width is $w=8 - 2x$ and the height is $h = x$. The volume $V(x)=(11 - 2x)(8 - 2x)x=4x^{3}-38x^{2}+88x$.

Step2: Differentiate the volume function

Using the power - rule $(x^n)'=nx^{n - 1}$, we have $V'(x)=12x^{2}-76x + 88$.

Step3: Set the derivative equal to zero

$12x^{2}-76x + 88 = 0$. Divide through by 4 to get $3x^{2}-19x + 22=0$.

Step4: Solve the quadratic equation

For a quadratic equation $ax^{2}+bx + c = 0$ ($a = 3$, $b=-19$, $c = 22$), the quadratic formula is $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. So $x=\frac{19\pm\sqrt{(-19)^{2}-4\times3\times22}}{2\times3}=\frac{19\pm\sqrt{361 - 264}}{6}=\frac{19\pm\sqrt{97}}{6}$.
$x_1=\frac{19+\sqrt{97}}{6}\approx\frac{19 + 9.85}{6}=\frac{28.85}{6}\approx4.81$ and $x_2=\frac{19-\sqrt{97}}{6}\approx\frac{19 - 9.85}{6}=\frac{9.15}{6}\approx1.53$.
We also need to consider the domain of $x$. Since $11-2x>0$ and $8 - 2x>0$, we have $x < 4$ and $x<5.5$, so we take $x\approx1.53\approx2$.
When $x = 2$, $V=(11-2\times2)(8 - 2\times2)\times2=(11 - 4)(8 - 4)\times2=7\times4\times2 = 56$.

Answer:

About 2 in. will create a box with a volume of about 56 in³.