Question

The maximum area of the rectangle that can be inscribed in a circle of radius 2 units is _____.

• A. $8\pi sq.units$
• B. $4sq.units$
• C. $5sq.units$
• D. $8sq.units$

Answer 1

The correct option is D $8sq.units$

Let the length of the rectangle be h and its breadth be k

Since the rectangle is inscribed inside the circle, its diagonal has to be the diameter of the circle.

$\therefore h^2+k^2=4^2...\left(1\right)$

The area of the rectangle is given by $hk$

Differentiating the condition w.r.t $h$,

$2h+2k\frac{dk}{dh}=0$

$\Rightarrow \frac{dk}{dh}=\frac{-h}{k}...\left(2\right)$

Now, differentiating the area since we need to maximize it,

$k+h\frac{dk}{dh}=0...\left(3\right)$

Substituting equation (2) into equation (3), we have

$k+h\times \frac{-h}{k}=0$

$\Rightarrow k^2=h^2$ or $k=h$

Substituting this relation in equation (1), we get

$2h^2=4^2$ or $h^2=8$

$\Rightarrow$ area $=hk=h^2=8$ sq unit