Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r.

Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r.

Answer

Question:
Find the dimensions of the rectangle oflargest area that can be inscribed in a circle of radius r.
LARGEST WILL BE A SQUARE WITH ITS DIAGONAL EQUAL TO DIAMETER OF THECIRCLE.(
DO YOU WANT A PROOF OF THIS STATEMENT?
IF SO PLEASE INFORM)
IF THE SIDE OF SQUARE IS X,HEN ITSDIAGONAL=X√2
HENCE
X
√2=DIAMETER = 2R
X=R√2
HENCE ANSWER IS A SQUARE WITH SIDE = R√2
ITS AREA = 2R^2
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PROOF OF STATEMENT MADE ABOVE
LET THE CENTER OF THE CIRCLE BE ORIGINAND 2 PERPENDICULAR DIAMETERS BE X AND Y AXES.SEE SKETCHBELOW.

LET THE RECTANGLE BE HKLM
LET COORDINATES OF E AND F BE (-A,0) AND (A,0)
HK=EF=LM
SINCE EQN . OF CIRCLE IS
X^2+Y^2=R^2
WE HAVE H AND K LYING ON THE CIRCLE AS
[-A,√(R^2-A^2)] AND[A,√(R^2-A^2)]
AREA = S = EF*KL=2A*2√(R^2-A^2)
DS/DA =0 FOR OPTIMUM
DS/DA=4[√(R^2-A^2)-A^2/√(R^2-A^2)]=0
(R^2-A^2)=A^2
R^2=2A^2
A=R/√2
HENCE
SIDE = 2A=2R/√2=R√2
OTHER SIDE = 2√(R^2-A^2)=2√[R^2-(R^2/2)]=2R/√2 =R√2
HENCE IT IS SQUARE WITH SIDE =R√2

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