a rectangular field has an area of $1,764\text{ m}^2$. the width of the field is $13\text{ m}$ more than the length. what is the perimeter of the field?

121 m
144 m
168 m
170 m

Question

a rectangular field has an area of $1,764\text{ m}^2$. the width of the field is $13\text{ m}$ more than the length. what is the perimeter of the field?

121 m
144 m
168 m
170 m

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Solving Quadratic Equations", "Quadratic Equation Formulation" ], "new_concepts": [], "current_concepts": [ "Solving Quadratic Equations", "Quadratic Equation Formulation" ] } ## Formulate the quadratic equation $$ \begin{aligned} &\text{Let } L \text{ be the length of the field in meters.}\ &\text{Width } W = L + 13\ &\text{Area } A = L \cdot W = L(L + 13) = 1764\ &L^2 + 13L - 1764 = 0 \end{aligned} $$ ## Solve for the length $$ \begin{aligned} &L = \frac{-13 \pm \sqrt{13^2 - 4(1)(-1764)}}{2}\ &L = \frac{-13 \pm \sqrt{169 + 7056}}{2}\ &L = \frac{-13 \pm \sqrt{7225}}{2}\ &L = \frac{-13 \pm 85}{2}\ &L = 36 \quad (L > 0) \end{aligned} $$ ## Calculate the perimeter $$ \begin{aligned} &W = 36 + 13 = 49\text{ m}\ &P = 2(L + W) = 2(36 + 49) = 2(85) = 170\text{ m} \end{aligned} $$

(A) 121 m

(B) 144 m

(C) 168 m

(D) 170 m

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

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QUESTION IMAGE

Question

Explanation:

Formulate the quadratic equation

Solve for the length

Calculate the perimeter

Answer: