find the dimensions of a rectangle whose width is 11 meters less than its length, and whose area is 242 square meters.

length =

width =

Question

find the dimensions of a rectangle whose width is 11 meters less than its length, and whose area is 242 square meters.

length =

width =

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Algebraic Translation", "Solving Quadratic Equations", "Quadratic Formula" ], "new_concepts": [], "current_concepts": [ "Algebraic Translation", "Solving Quadratic Equations", "Quadratic Formula" ] } ## Set up the equations $$ \begin{aligned} &\text{Let } L = \text{length (in meters)}\ &\text{Let } W = \text{width (in meters)}\ &W = L - 11\ &\text{Area } = L \times W = 242\ &L(L - 11) = 242\ &L^2 - 11L - 242 = 0 \end{aligned} $$ ## Solve the quadratic equation $$ \begin{aligned} &L = \frac{-(-11) \pm \sqrt{(-11)^2 - 4(1)(-242)}}{2(1)}\ &L = \frac{11 \pm \sqrt{121 + 968}}{2}\ &L = \frac{11 \pm \sqrt{1089}}{2}\ &L = \frac{11 \pm 33}{2}\ &L = 22 \quad (\text{since } L > 0) \end{aligned} $$ ## Calculate the width $$ \begin{aligned} &W = L - 11\ &W = 22 - 11 = 11 \end{aligned} $$ Length = 22 meters Width = 11 meters { "subject": "Mathematics", "question_type": "Word Problem", "knowledge_point": [ "Mathematics", "Algebra", "Solving Quadratic Equations" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Set up the equations

Solve the quadratic equation

Calculate the width

Answer: