Assertion:  The rate of change of area of a circle with respect to its radius r when r= 6 cm is 12$\pi$ cm².

            Reason:  Rate of change of area of a circle with respect to its radius r is $\frac{dA}{dr}$, where A is the area of the circle.

Let assume that we have rate of change of the area of a circle with respect to it's radius r when $r=6\mathrm{cm}$.

As we know that, area of circle is given by: $\mathrm{\pi }\cdot {\mathrm{r}}^2$

where r is the radius of the circle.

Let $\mathrm{A}=\mathrm{\pi }\cdot {\mathrm{r}}^2$

As we know that, if  $\mathrm{y}=\mathrm{f}\left(\mathrm{x}\right)$,  then dy/dx denotes the rate of change of y with respect to x.

So, by differentiating A with respect to r we get,

$\Rightarrow \frac{\mathrm{dA}}{\mathrm{dr}}=\frac{\mathrm{d}\left(\mathrm{\pi }\cdot {\mathrm{r}}^2\right)}{\mathrm{dr}}=2\mathrm{\pi r}$

Now we have to find the value of dA/dr at r = 6cm i.e  ${\left[\frac{dA}{dr}\right]}_{r=6}$

$\Rightarrow {\left[\frac{\mathrm{dA}}{\mathrm{dr}}\right]}_{\mathrm{r}=6}=2\mathrm{\pi }\cdot 6=12\mathrm{\pi cm}$

Hence, the rate of change of the area of a circle with respect to its radius r when r=6cm is 12$\pi$cm.

Therefore, both A and R are true and R is the correct explanation of A.