the length of a rectangle is 4 more than the width. the area is 117 square feet. find the length and width of the rectangle.

answer
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width = feet
length = feet

Question

the length of a rectangle is 4 more than the width. the area is 117 square feet. find the length and width of the rectangle.

answer
how to enter your answer (opens in new window)
width =   feet
length =   feet

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Geometric Quadratic Modeling", "Solving Quadratic Equations" ], "new_concepts": [], "current_concepts": [ "Geometric Quadratic Modeling", "Solving Quadratic Equations", "Factoring Quadratic Equations" ] } ## Define variables and set up the area equation Using the *Geometric Quadratic Modeling* knowledge point $$ \begin{aligned} &\text{Let } w \text{ be the width of the rectangle in feet.}\ &\text{Length } l = w + 4\ &\text{Area } A = w \cdot l = w(w + 4) = 117 \end{aligned} $$ ## Write the quadratic equation in standard form Using the *Standard Form of Quadratic Equations* knowledge point $$ \begin{aligned} &w^2 + 4w = 117\ &w^2 + 4w - 117 = 0 \end{aligned} $$ ## Solve the quadratic equation by factoring Using the *Factoring Quadratic Equations* and *Zero Product Property* knowledge points $$ \begin{aligned} &(w + 13)(w - 9) = 0\ &w = -13 \quad \text{or} \quad w = 9 \end{aligned} $$ ## Determine the dimensions of the rectangle Using the *Geometric Quadratic Modeling* knowledge point $$ \begin{aligned} &\text{Since width must be positive, } w = 9\text{ feet.}\ &l = w + 4 = 9 + 4 = 13\text{ feet.} \end{aligned} $$ The length of a rectangle is 4 more than the width. The area is 117 square feet. Find the length and width of the rectangle. Width = 9 feet Length = 13 feet { "elements": [ { "type": "rectangle", "params": [[-4.5, -3], [4.5, 3]], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3, "fillColor": "rgba(140, 85, 242, 0.15)", "name": "Rectangle" } }, { "type": "text", "params": [0, -3.5, "$$w = 9\text{ ft}$$"], "properties": { "anchorX": "middle", "anchorY": "top", "color": "#583C87" } }, { "type": "text", "params": [4.8, 0, "$$l = w + 4 = 13\text{ ft}$$"], "properties": { "anchorX": "left", "anchorY": "middle", "color": "#583C87" } }, { "type": "text", "params": [0, 0, "$$\text{Area} = 117\text{ ft}^2$$"], "properties": { "anchorX": "middle", "anchorY": "middle", "color": "#8C55F2" } } ] } { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Geometric Quadratic Modeling" ] }

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QUESTION IMAGE

Question

Explanation:

Define variables and set up the area equation

Write the quadratic equation in standard form

Solve the quadratic equation by factoring

Determine the dimensions of the rectangle

Answer: