the volume of the cereal box must be 192 \text{ in.}^3.

height: $(x + 10)$ \text{ in.}
width: $(x - 4)$ \text{ in.}
length: $x$ \text{ in.}

polynomial equation: $x^3 + 6x^2 - 40x = 192$
solutions: $x = -8$, $x = -4$, and $x = 6$

in the context of this problem, which solutions to the polynomial equation can you eliminate because they do not make sense?
$x = -8$
$x = -4$
$x = 6$

what are the dimensions of the cereal box?
the length is in., the width is in., and the height is in.

Question

the volume of the cereal box must be 192 \text{ in.}^3.

height: $(x + 10)$ \text{ in.}
width: $(x - 4)$ \text{ in.}
length: $x$ \text{ in.}

polynomial equation: $x^3 + 6x^2 - 40x = 192$
solutions: $x = -8$, $x = -4$, and $x = 6$

in the context of this problem, which solutions to the polynomial equation can you eliminate because they do not make sense?
$x = -8$
$x = -4$
$x = 6$

what are the dimensions of the cereal box?
the length is   in., the width is   in., and the height is   in.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Polynomial Word Problems" ], "new_concepts": [], "current_concepts": [ "Polynomial Word Problems", "Extraneous Solutions" ] } ## Evaluate physical constraints on dimensions $$ \begin{aligned} &\text{Length: } x > 0 \ &\text{Width: } x - 4 > 0 \implies x > 4 \ &\text{Height: } x + 10 > 0 \implies x > -10 \end{aligned} $$ ## Determine valid solution $$ \begin{aligned} &x \in \{-8, -4, 6\} \ &x > 4 \implies x = 6 \end{aligned} $$ ## Calculate dimensions $$ \begin{aligned} &\text{Length} = x = 6 \ &\text{Width} = x - 4 = 6 - 4 = 2 \ &\text{Height} = x + 10 = 6 + 10 = 16 \end{aligned} $$ What are the dimensions of the cereal box? The length is 6 in., the width is 2 in., and the height is 16 in. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Polynomial Word Problems" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Evaluate physical constraints on dimensions

Determine valid solution

Calculate dimensions

Answer: