an architect wants to draw a rectangle with a diagonal of 15 millimeters. the length of the rectangle is to be 6 millimeters more than twice the width. what dimensions should she make the rectangle?

answer
length = millimeters
width = millimeters

Question

an architect wants to draw a rectangle with a diagonal of 15 millimeters. the length of the rectangle is to be 6 millimeters more than twice the width. what dimensions should she make the rectangle?

answer
length =   millimeters
width =   millimeters

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Translating Word Problems to Quadratic Equations", "Pythagorean Theorem Applications", "Solving Quadratic Equations" ], "new_concepts": [], "current_concepts": [ "Translating Word Problems to Quadratic Equations", "Pythagorean Theorem Applications", "Solving Quadratic Equations" ] } ## Set up the variables Using the *Translating Word Problems to Quadratic Equations* knowledge point Let $w$ be the width of the rectangle in millimeters. The length $l$ is $6$ millimeters more than twice the width: $$l = 2w + 6$$ The diagonal $d$ is $15$ millimeters. ## Apply the Pythagorean theorem Using the *Pythagorean Theorem Applications* knowledge point $$ \begin{aligned} w^2 + l^2 &= d^2 \ w^2 + (2w + 6)^2 &= 15^2 \end{aligned} $$ ## Expand and simplify the equation Using the *Solving Quadratic Equations* knowledge point $$ \begin{aligned} w^2 + 4w^2 + 24w + 36 &= 225 \ 5w^2 + 24w - 189 &= 0 \end{aligned} $$ ## Solve the quadratic equation Using the *Solving Quadratic Equations* knowledge point $$ \begin{aligned} w &= \frac{-24 \pm \sqrt{24^2 - 4(5)(-189)}}{2(5)} \ w &= \frac{-24 \pm \sqrt{576 + 3780}}{10} \ w &= \frac{-24 \pm \sqrt{4356}}{10} \ w &= \frac{-24 \pm 66}{10} \end{aligned} $$ Since width must be positive: $$ \begin{aligned} w &= \frac{-24 + 66}{10} = \frac{42}{10} = 4.2 \end{aligned} $$ ## Calculate the length Using the *Translating Word Problems to Quadratic Equations* knowledge point $$ \begin{aligned} l &= 2(4.2) + 6 \ l &= 8.4 + 6 = 14.4 \end{aligned} $$ Length = 14.4 millimeters Width = 4.2 millimeters { "elements": [ { "type": "point", "params": [[0, 0]], "properties": { "name": "A", "color": "#583C87", "withLabel": false } }, { "type": "point", "params": [[7.2, 0]], "properties": { "name": "B", "color": "#583C87", "withLabel": false } }, { "type": "point", "params": [[7.2, 2.1]], "properties": { "name": "C", "color": "#583C87", "withLabel": false } }, { "type": "point", "params": [[0, 2.1]], "properties": { "name": "D", "color": "#583C87", "withLabel": false } }, { "type": "polygon", "params": [[0, 0], [7.2, 0], [7.2, 2.1], [0, 2.1]], "properties": { "fillColor": "rgba(140, 85, 242, 0.15)", "strokeColor": "#8C55F2", "strokeWidth": 2 } }, { "type": "segment", "params": [[0, 0], [7.2, 2.1]], "properties": { "strokeColor": "#F2557F", "strokeWidth": 2, "dash": 2 } }, { "type": "text", "params": [[3.6, -0.5], "Length = 14.4 mm"] }, { "type": "text", "params": [[7.5, 1.05], "Width = 4.2 mm"] }, { "type": "text", "params": [[3.0, 1.3], "15 mm"] } ] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Translating Word Problems to Quadratic Equations" ] }

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QUESTION IMAGE

Question

Explanation:

Set up the variables

Apply the Pythagorean theorem

Expand and simplify the equation

Solve the quadratic equation

Calculate the length

Answer: