tommy has a rectangular-shaped swimming pool. he wants to know the volume of water in the pool, but he is not sure what the width of the pool is. he knows the relationship between the width and the length and height of the pool. he used this relationship to come up with the following equation to find the volume of the pool when the width is $w$ meters.

\v(w) = w^3 + 6w^2 - 16w\

using the end behavior of $v(w)$, determine the graph of the function.

Question

tommy has a rectangular-shaped swimming pool. he wants to know the volume of water in the pool, but he is not sure what the width of the pool is. he knows the relationship between the width and the length and height of the pool. he used this relationship to come up with the following equation to find the volume of the pool when the width is \$w\$ meters.

\\v(w) = w^3 + 6w^2 - 16w\\

using the end behavior of \$v(w)\$, determine the graph of the function.

\v(w) = w^3 + 6w^2 - 16w\

using the end behavior of $v(w)$, determine the graph of the function.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Cubic Functions", "End Behavior", "Zeros of Polynomial Functions" ], "new_concepts": [], "current_concepts": [ "Cubic Functions", "End Behavior", "Zeros of Polynomial Functions", "Graphing Polynomials" ] } ## Analyze the degree and leading coefficient $$ \begin{aligned} &V(w) = w^3 + 6w^2 - 16w\ &\text{Degree } (n) = 3 \quad (\text{odd})\ &\text{Leading Coefficient } (a_n) = 1 > 0 \quad (\text{positive}) \end{aligned} $$ ## Determine the end behavior $$ \begin{aligned} &\lim_{w \to \infty} V(w) = \infty\ &\lim_{w \to -\infty} V(w) = -\infty \end{aligned} $$ ## Find the zeros of the function $$ \begin{aligned} &w^3 + 6w^2 - 16w = 0\ &w(w^2 + 6w - 16) = 0\ &w(w + 8)(w - 2) = 0\ &w = -8, \quad w = 0, \quad w = 2 \end{aligned} $$

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{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Graphing Polynomials" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze the degree and leading coefficient

Determine the end behavior

Find the zeros of the function

Answer: