a new rectangular vegetable garden is to be completely enclosed by 40 m of fencing. what dimensions can the garden be, if the area of the garden must be 96 \text{ m}^2?

- length = 14 m and width = 6 m
- length = 11 m and width = 9 m
- length = 12 m and width = 8 m
- length = 13 m and width = 7 m

Question

a new rectangular vegetable garden is to be completely enclosed by 40 m of fencing. what dimensions can the garden be, if the area of the garden must be 96 \text{ m}^2?

- length = 14 m and width = 6 m
- length = 11 m and width = 9 m
- length = 12 m and width = 8 m
- length = 13 m and width = 7 m

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Quadratic Word Problems" ], "new_concepts": [], "current_concepts": [ "Quadratic Word Problems", "Linear Relations" ] } ## Set up the system of equations $$ \begin{aligned} &2(l + w) = 40 \implies l + w = 20 \ &l \cdot w = 96 \end{aligned} $$ ## Solve the quadratic equation $$ \begin{aligned} &w(20 - w) = 96 \ &w^2 - 20w + 96 = 0 \ &(w - 12)(w - 8) = 0 \ &w = 12 \text{ or } w = 8 \end{aligned} $$ ## Determine the dimensions $$ \begin{aligned} &\text{length} = 12\text{ m} \ &\text{width} = 8\text{ m} \end{aligned} $$

length = 14 m and width = 6 m

length = 11 m and width = 9 m

length = 12 m and width = 8 m

length = 13 m and width = 7 m

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

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Set up the system of equations

Solve the quadratic equation

Determine the dimensions

Answer: