a rectangular deck is 12 ft by 14 ft. when the length and width are increased by the same amount, the area becomes 288 sq. ft. by how much were the dimensions increased?

2 ft
4 ft
8 ft
10 ft

Question

a rectangular deck is 12 ft by 14 ft. when the length and width are increased by the same amount, the area becomes 288 sq. ft. by how much were the dimensions increased?

2 ft
4 ft
8 ft
10 ft

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Quadratic Equation Formulation", "Solving Quadratic Equations" ], "new_concepts": [], "current_concepts": [ "Quadratic Equation Formulation", "Solving Quadratic Equations" ] } ## Formulate the quadratic equation $$ \begin{aligned} &\text{Let } x \text{ be the amount by which each dimension is increased.}\ &(12 + x)(14 + x) = 288\ &x^2 + 26x + 168 = 288\ &x^2 + 26x - 120 = 0 \end{aligned} $$ ## Solve the quadratic equation $$ \begin{aligned} &(x + 30)(x - 4) = 0\ &x = -30 \quad \text{or} \quad x = 4 \end{aligned} $$ ## Determine the valid dimension increase $$ \begin{aligned} &x > 0 \implies x = 4\text{ ft} \end{aligned} $$

(A) 2 ft

(B) 4 ft

(C) 8 ft

(D) 10 ft

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

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Question

Explanation:

Formulate the quadratic equation

Solve the quadratic equation

Determine the valid dimension increase

Answer: