Answer:
2r2units
Explanation:
Let a rectangle of dimensions x×y
Be inscribed in a circle of radius r.
Applying Pythagoras theorem on the right-angled triangle,
⇒x2+y2=(2r)2⇒x2+y2=4r2⇒y2=4r2−x2⇒y=4r2−x2−−−−−−−√
As dimensions of any shape cannot be negative so y=−4r2−x2−−−−−−−√
is not considered.
Area of the rectangle,
⇒A=x×y⇒A=x4r2−x2−−−−−−−√
Differentiating both sides with respect to x, we get –
dAdx=ddx(x4r2−x2−−−−−−−√)
Use the product rule to differentiate the above result –
A1(x)=xd(4r2−x2−−−−−−−√)dx+4r2−x2−−−−−−−√d(x)dxA1(x)=xd4r2−x2−−−−−−−√d(4r2−x2)d(4r2−x2)dx+4r2−x2−−−−−−−√(1)A1(x)=x124r2−x2−−−−−−−√(−2x)+4r2−x2−−−−−−−√A1(x)=−x2+4r2−x24r2−x2−−−−−−−√A1(x)=−2x2+4r24r2−x2−−−−−−−√
Now to get the maximum value, put A1(x)=0
−2x2+4r24r2−x2−−−−−−−√=0⇒4r2=2x2x2=2r2x=2–√r
The negative value of x is rejected.
⇒y=4r2−x2−−−−−−−√=r2–√
A(r2–√)=r2–√(4r2−2r2−−−−−−−√A(r2–√)=r2–√(r2–√)=2r2
The maximum area of a rectangle inscribed in a circle is 2r2units
and its dimensions are r2–√units×r2–√units
So, the correct answer is “ 2r2units ”.