a rectangle is inscribed with its base on the $x$-axis and its upper corners on the parabola $y = 7 - x^2$. what are the dimensions of such a rectangle with the greatest possible area?
round to two decimal places.

width =
height =

Question

a rectangle is inscribed with its base on the $x$-axis and its upper corners on the parabola $y = 7 - x^2$. what are the dimensions of such a rectangle with the greatest possible area?
round to two decimal places.

width =
height =

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Optimization Problems", "Critical Points" ], "new_concepts": [], "current_concepts": [ "Optimization Problems", "Quadratic Optimization", "Critical Points" ] } ## Formulate the area function $$ \begin{aligned} &\text{Let the upper right corner of the rectangle be } (x, y) \text{ on the parabola } y = 7 - x^2 \text{ with } x > 0.\ &\text{Width } = 2x\ &\text{Height } = y = 7 - x^2\ &A(x) = \text{Width} \times \text{Height} = 2x(7 - x^2) = 14x - 2x^3 \end{aligned} $$ ## Find the critical point $$ \begin{aligned} &A'(x) = 14 - 6x^2 = 0\ &x^2 = \frac{14}{6} = \frac{7}{3}\ &x = \sqrt{\frac{7}{3}} \approx 1.5275 \end{aligned} $$ ## Calculate optimal dimensions $$ \begin{aligned} &\text{Width} = 2x = 2\sqrt{\frac{7}{3}} \approx 3.055 \approx 3.06\ &\text{Height} = 7 - x^2 = 7 - \frac{7}{3} = \frac{14}{3} \approx 4.666 \approx 4.67 \end{aligned} $$ Width = 3.06 Height = 4.67 { "elements": [ { "type": "functiongraph", "params": [ { "js": "7 - x*x", "latex": "y = 7 - x^2" }, -3.5, 3.5 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2, "name": "y = 7 - x^2", "withLabel": true } }, { "type": "polygon", "params": [ [-1.5275, 0], [1.5275, 0], [1.5275, 4.6667], [-1.5275, 4.6667] ], "properties": { "fillColor": "rgba(85, 131, 242, 0.3)", "strokeColor": "#5583F2", "strokeWidth": 2 } }, { "type": "point", "params": [ [1.5275, 4.6667] ], "properties": { "name": "(x, 7-x^2)", "color": "#F2557F", "size": 4, "withLabel": true } } ] } { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Optimization Problems" ] }

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QUESTION IMAGE

Question

Explanation:

Formulate the area function

Find the critical point

Calculate optimal dimensions

Answer: