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=4r
2
sinθcosθ=2r
2
sin2θ
⇒A(θ)=2r
2
sin2θ
⇒A
′
(θ)=4r
2
cos2θ
⇒A”(θ)=−8r
2
sin2θ
⇒A
′
(θ)=0
⇒4r
2
cos2θ=0
⇒cos2θ=0⇒2θ=
2
π
⇒
4
π
Now A
′′
(
4
π
)=−8r
2
×1=−8r
2
<0
A is largest when θ=
4
π
When θ=
4
π
2x=2rcos×
4
π
=2r(
2
1
)=
2r
2y=2rsin
4
π
=2r(
2
1
)=
2r
∴ the dimensions of the rectangle are
2
r and
2
r.
(The rectangle is also square)