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