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Question

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

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Solution

Let us take the circle with centre (0,0) and radius r and PQRS be the rectangle inscribed in the circle.
Let x=rcosθ;y=rsinθ
The dimensions of the rectangle are
2x=2rcosθ;2y=2rsinθ
Area at rectangle A=4r2sinθcosθ=2r2sin2θ
A(θ)=2r2sin2θ
A(θ)=4r2cos2θ
A"(θ)=8r2sin2θ
A(θ)=0
4r2cos2θ=0
cos2θ=02θ=π2π4
Now A′′(π4)=8r2×1=8r2<0
A is largest when θ=π4
When θ=π4
2x=2rcos×π4=2r(12)=2r
2y=2rsinπ4=2r(12)=2r
the dimensions of the rectangle are 2r and 2r.
(The rectangle is also square)
633216_608498_ans.png

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