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{ "quality": "incomplete", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Quadratic Optimization", "Vertex of Parabola" ], "current_concepts": [ "Quadratic Optimization", "Vertex of Parabola" ] } ## Identify the quadratic model and optimization goal $$ \begin{aligned} &S(x) = -2x^2 + 80x + 300\ &a = -2,\quad b = 80,\quad c = 300 \end{aligned} $$ ## Calculate the optimal advertising spend $$ \begin{aligned} &x_{\text{opt}} = -\frac{b}{2a} = -\frac{80}{2(-2)} = 20 \end{aligned} $$ ## Calculate the maximum number of sign-ups $$ \begin{aligned} &S(20) = -2(20)^2 + 80(20) + 300\ &S(20) = -2(400) + 1600 + 300\ &S(20) = -800 + 1600 + 300 = 1100 \end{aligned} $$ The optimal advertising spend is $20$ thousand dollars (which is $\$20,000$), resulting in a maximum of $1,100$ customer sign-ups. { "subject": "Mathematics", "question_type": "Word Problem", "knowledge_point": [ "Mathematics", "Algebra", "Quadratic Optimization" ] }

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Explanation:

Identify the quadratic model and optimization goal

Calculate the optimal advertising spend

Calculate the maximum number of sign-ups

Answer: