Asked by Kris
what are the dimensions of the rectangle of largest aera that can be inscribed under the curve y=8-x^2 and above the x-axis
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GPT-5 mini
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Take the rectangle symmetric about the y-axis with top corners at (±x, y). Then y = 8 − x^2 and width = 2x, so
A(x) = 2x(8 − x^2) = 16x − 2x^3, 0 ≤ x ≤ √8.
Differentiate: A'(x) = 16 − 6x^2 = 0 ⇒ x^2 = 8/3 ⇒ x = √(8/3).
Height y = 8 − x^2 = 16/3. Width = 2x = 2√(8/3) = (4√6)/3.
Thus the largest-area rectangle has width (4√6)/3 and height 16/3 (top corners at (±√(8/3), 16/3)). The maximum area is A = (64√6)/9 ≈ 17.4186.
A(x) = 2x(8 − x^2) = 16x − 2x^3, 0 ≤ x ≤ √8.
Differentiate: A'(x) = 16 − 6x^2 = 0 ⇒ x^2 = 8/3 ⇒ x = √(8/3).
Height y = 8 − x^2 = 16/3. Width = 2x = 2√(8/3) = (4√6)/3.
Thus the largest-area rectangle has width (4√6)/3 and height 16/3 (top corners at (±√(8/3), 16/3)). The maximum area is A = (64√6)/9 ≈ 17.4186.
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