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A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y = 2-x^2. What are the dimensions of such a rectangle with the great

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1 Answer

Coordinates of upper right corner = (x, 2-x2)
 
Length of base = 2x
Height = 2-x2
 
A = area of rectangle = 2x(2 - x2) = 4x - 2x3 , 0 < x < √2
 
A' = 4 - 6x2
 
A' = 0 when x = √(2/3)
 
When 0 < x < √(2/3), A' > 0.  So, A is increasing
 
When √(2/3) < x < √2, A' < 0.  So, A is decreasing
 
Maximum area when x = √(2/3)
 
Dimensions:  length = 2x = 2√(2/3)
 
                    height = 2 - x= 4/3