I would stick with x^2+y^2 =10^2 = 100
then the area = 2x*2y = 4 x y
A = 4 x y where y = (100-x^2)^.5
A = 4 x (100-x^2)^.5
dA/dx = 0 = 4 [-2x^2 (.5)(100-x^2)^-.5 +(100-x^2)^.5 ]
x^2 = 100 - x^2
2 x^2 = 100
x^2 = 50
x = about 7
y = about 7
area = about 4*49 = 196
You can do this all more carefully and accurately
Find the dimensions of a rectangle with maximum area that can be inscribed in a circle of a radius of 10.
Okay, so I know that I am going to need the Pythagorean theorem, where x^2+y^2=20^2 (20 is from the doubling of the radius which actually makes the hypotenuse of the rectangle.)
Up to there, I have no clue. I just know it has something involved with derivatives.
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