The rate of change of area of a circle with respect to its circumference when radius in 6 cm, is

$6\, cm^2/cm$

The correct answer is Option (3) → $6\, cm^2/cm$

Let:

Area of circle: $A = \pi r^2$

Circumference: $C = 2\pi r$

Step 1: Express $A$ in terms of $C$

From $C = 2\pi r \Rightarrow r = \frac{C}{2\pi}$

Substitute into $A$:

$A = \pi \left( \frac{C}{2\pi} \right)^2 = \pi \cdot \frac{C^2}{4\pi^2} = \frac{C^2}{4\pi}$

Step 2: Differentiate $A$ with respect to $C$

$\frac{dA}{dC} = \frac{1}{4\pi} \cdot 2C = \frac{C}{2\pi}$

Step 3: Use $r = 6$ cm to find $C$

$C = 2\pi r = 12\pi$

Then, $\frac{dA}{dC} = \frac{12\pi}{2\pi} = 6$