question 8 (8 points)

a farmer wants to build a rectangular pen using 120 feet of fencing. one side of the pen is along a barn, so fencing is needed for only three sides.

find the dimensions of the pen that maximize the area, and find the maximum area.

show your setup and work.

Question

question 8 (8 points)

a farmer wants to build a rectangular pen using 120 feet of fencing. one side of the pen is along a barn, so fencing is needed for only three sides.

find the dimensions of the pen that maximize the area, and find the maximum area.

show your setup and work.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Optimization Problems", "Geometric Optimization" ], "new_concepts": [], "current_concepts": [ "Optimization Problems", "Geometric Optimization", "Domain of Optimization" ] } ## Set up the variables and objective function Using the *Geometric Optimization* knowledge point $$ \begin{aligned} &\text{Let } x \text{ be the length of the two sides perpendicular to the barn (in feet).}\ &\text{Let } y \text{ be the length of the side parallel to the barn (in feet).}\ &\text{Constraint: } 2x + y = 120 \implies y = 120 - 2x\ &\text{Objective Area Function: } A(x) = x \cdot y = x(120 - 2x) = 120x - 2x^2 \end{aligned} $$ ## Determine the domain of the function Using the *Domain of Optimization* knowledge point $$ \begin{aligned} &x \ge 0\ &y \ge 0 \implies 120 - 2x \ge 0 \implies x \le 60\ &\text{Domain: } x \in [0, 60] \end{aligned} $$ ## Find the critical points Using the *Optimization Problems* knowledge point $$ \begin{aligned} &A'(x) = 120 - 4x\ &A'(x) = 0 \implies 120 - 4x = 0 \implies x = 30 \end{aligned} $$ ## Evaluate the extrema and find the dimensions Using the *Absolute Extrema on Closed Intervals* knowledge point $$ \begin{aligned} &A(0) = 0\ &A(60) = 0\ &A(30) = 30(120 - 2(30)) = 30(60) = 1800\ &\text{Maximum Area: } 1800\text{ sq ft}\ &\text{Dimensions: } x = 30\text{ ft},\ y = 60\text{ ft} \end{aligned} $$ * **Dimensions of the pen:** 30 feet (perpendicular to the barn) by 60 feet (parallel to the barn) * **Maximum area:** 1,800 square feet { "elements": [ { "type": "line", "params": [[-2, 4], [8, 4]], "properties": { "strokeColor": "#583C87", "strokeWidth": 4, "name": "Barn Wall" } }, { "type": "segment", "params": [[0, 4], [0, 0]], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3, "name": "x" } }, { "type": "segment", "params": [[6, 4], [6, 0]], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3, "name": "x" } }, { "type": "segment", "params": [[0, 0], [6, 0]], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3, "name": "y" } }, { "type": "text", "params": [3, 4.3, "Barn Wall"], "properties": { "color": "#583C87" } }, { "type": "text", "params": [-0.5, 2, "x = 30 ft"], "properties": { "color": "#8C55F2" } }, { "type": "text", "params": [6.5, 2, "x = 30 ft"], "properties": { "color": "#8C55F2" } }, { "type": "text", "params": [3, -0.5, "y = 60 ft"], "properties": { "color": "#8C55F2" } } ], "timestamps": [0.1] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Calculus", "Optimization Problems" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Set up the variables and objective function

Determine the domain of the function

Find the critical points

Evaluate the extrema and find the dimensions

Answer: