the area of the conference table in mr. nathans office must be no more than $175\text{ ft}^2$. if the length of the table is $18\text{ ft}$ more than the width, $x$, which interval can be the possible widths?

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Question

the area of the conference table in mr. nathans office must be no more than \$175\text{ ft}^2\$. if the length of the table is \$18\text{ ft}\$ more than the width, \$x\$, which interval can be the possible widths?

\$0 < x \le 25\$
\$0 < x \le 7\$
\$0 \le x \le 7\$
\$0 \le x \le 25\$

$0 < x \le 25$
$0 < x \le 7$
$0 \le x \le 7$
$0 \le x \le 25$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Quadratic Inequalities", "Solving Quadratic Equations" ], "new_concepts": [], "current_concepts": [ "Quadratic Inequalities", "Solving Quadratic Equations", "Quadratic Modeling Interpretation" ] } ## Formulate the inequality $$ \begin{aligned} &\text{Width} = x\ &\text{Length} = x + 18\ &\text{Area} = x(x + 18) \le 175\ &x^2 + 18x - 175 \le 0 \end{aligned} $$ ## Solve the quadratic equation $$ \begin{aligned} &x^2 + 18x - 175 = 0\ &(x + 25)(x - 7) = 0\ &x = -25 \quad \text{or} \quad x = 7 \end{aligned} $$ ## Determine the physical domain $$ \begin{aligned} &-25 \le x \le 7\ &x > 0 \quad (\text{since width must be positive})\ &0 < x \le 7 \end{aligned} $$

(A) $0 < x \le 25$

(B) $0 < x \le 7$

(C) $0 \le x \le 7$

(D) $0 \le x \le 25$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Quadratic Inequalities" ] }

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QUESTION IMAGE

Question

Explanation:

Formulate the inequality

Solve the quadratic equation

Determine the physical domain

Answer: