Question

spiral review • assessment readiness

1. the volume v of a gift box (in cubic inches) is modeled by the function $v(w) = w^3 + 3w^2 - 10w$ where $w$ is the width (in inches). what is the width of a gift box with volume 264 cubic inches?

$\boldsymbol{\text{a}}$ 6 inches
$\boldsymbol{\text{b}}$ 7 inches
$\boldsymbol{\text{c}}$ 8 inches
$\boldsymbol{\text{d}}$ 9 inches

1. for the functions $f(x)$ and $g(x)$, $f(x) = 3x - 5$ and $g(x) = 2x + 1$ the function $h(x) = 3\cdot g(x) - 5$. which equation is equivalent to $h(x)$?

$\boldsymbol{\text{a}}$ $y = 6x + 1$
$\boldsymbol{\text{b}}$ $y = 6x + 3$
$\boldsymbol{\text{c}}$ $y = 6x - 2$
$\boldsymbol{\text{d}}$ $y = 6x - 8$

1. match equivalent expressions.

$\boldsymbol{\text{a.}}$ $sqrt{(xy)^3}$ $\boldsymbol{\text{1.}}$ $(xy)^{\frac{2}{3}}$
$\boldsymbol{\text{b.}}$ $sqrt4{xy}$ $\boldsymbol{\text{2.}}$ $(xy)^{\frac{1}{4}}$
$\boldsymbol{\text{c.}}$ $sqrt3{(xy)^4}$ $\boldsymbol{\text{3.}}$ $(xy)^{\frac{3}{2}}$
$\boldsymbol{\text{d.}}$ $(sqrt3{xy})^2$ $\boldsymbol{\text{4.}}$ $(xy)^{\frac{1}{3}}$
$\boldsymbol{\text{e.}}$ $sqrt3{xy}$ $\boldsymbol{\text{5.}}$ $(xy)^{\frac{4}{3}}$

1. which is a zero of the function $p(x) = x^3 - x^2 + 4x - 4$?

$\boldsymbol{\text{a}}$ 2
$\boldsymbol{\text{b}}$ 1
$\boldsymbol{\text{c}}$ 0
$\boldsymbol{\text{d}}$ $-2$

Explanation:

Question 63

Step1: Set volume equal to function

$V(w) = 264 = w^3 + 3w^2 - 10w$

Step2: Rearrange to standard polynomial

$w^3 + 3w^2 - 10w - 264 = 0$

Step3: Test integer solutions (Rational Root Theorem)

Test $w=6$: $6^3 + 3(6)^2 -10(6)-264=216+108-60-264=0$

Step1: Substitute $g(x)$ into $h(x)$

$h(x)=3(2x+1)-5$

Step2: Distribute the 3

$h(x)=6x+3-5$

Step3: Combine constant terms

$h(x)=6x-2$

Use exponent rules: $\sqrt[n]{a^m}=a^{\frac{m}{n}}$ and $(a^m)^n=a^{mn}$

• A. $\sqrt{(xy)^3}=(xy)^{\frac{3}{2}}$ → matches 3
• B. $\sqrt[4]{xy}=(xy)^{\frac{1}{4}}$ → matches 2
• C. $\sqrt[3]{(xy)^4}=(xy)^{\frac{4}{3}}$ → matches 5
• D. $(\sqrt[3]{xy})^2=(xy)^{\frac{2}{3}}$ → matches 1
• E. $\sqrt[3]{xy}=(xy)^{\frac{1}{3}}$ → matches 4

Answer:

A 6 inches

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Question 64