Question

alg 2 do now:
①
find the volume of
the finished box
② factor and graph
y = x⁴ - 3x³
③ sketch y = 2(x + 1)³

Explanation:

For Question ①:

Step1: Define box dimensions

Let cutout side = $x$.
Length: $10-2x$, Width: $8-2x$, Height: $x$

Step2: Volume formula for box

Volume $V = \text{length} \times \text{width} \times \text{height}$
$V(x) = x(10-2x)(8-2x)$

Step3: Expand the expression

First multiply $(10-2x)(8-2x) = 80-20x-16x+4x^2 = 4x^2-36x+80$
Then multiply by $x$: $V(x) = 4x^3-36x^2+80x$

Step1: Factor the polynomial

Factor out greatest common factor $x^3$
$y = x^3(x - 3)$

Step2: Find roots for graph

Set $y=0$: $x^3(x-3)=0$ → Roots at $x=0$ (multiplicity 3) and $x=3$ (multiplicity 1)

Step3: Analyze end behavior

As $x\to+\infty$, $y\to+\infty$; as $x\to-\infty$, $y\to-\infty$

Step4: Describe graph shape

Crosses x-axis at $x=3$, touches and flattens at $x=0$ (odd multiplicity 3)

Step1: Identify parent function

Parent function: $y = x^3$

Step2: Identify transformations

Horizontal shift: left 1 unit (from $(x+1)$)
Vertical stretch: by factor of 2 (from coefficient 2)

Step3: Key point transformations

Parent key point $(0,0)$ → $(-1, 0)$; $(1,1)$ → $(0, 2)$; $(-1,-1)$ → $(-2, -2)$

Step4: Sketch curve

Draw cubic curve through transformed points, stretched vertically, shifted left 1

Answer:

$V(x) = 4x^3 - 36x^2 + 80x$ (where $0 < x < 4$ to keep dimensions positive)

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For Question ②: