SOLUTION: A rectangle is inscribed in a circle: x^2 + y^2 = 9. Find the dimensions of the rectangle with the maximum area.

Tutor greenestamps above assumed that a rectangle of maximum area inscribed in
a circle is a square.  If you are not a calculus student, and/or your teacher
says that can be assumed, indeed his solution is acceptable.  Below I have
shown the solution in case that assumption is not permitted.



The area of the rectangle is given by A = (2x)(2y) = 4xy
We want to show that the rectangle is a square, that 
the length = 2x = the width 2y, or x = y and, as greenestamps 
essentially stated:







Rationalizing the denominator


So the length and with are both 2x = 2y = 

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However in case you are required to prove that the rectangle is a square:
 
We will take the point (x,y) in QI so x and y are both positive.



We want to show that x = y so that the dimensions, 2x by 2y are the same.

Squaring both sides and differentiating explicitly avoids messy 
square roots and fraction exponents:






Since A = 4xy, divide the left side by A and the right side by 4xy



We need 







Substituting,



We set that equal to 0





Since x and y are both positive


That is what we needed to prove.

Edwin