If a circular plate is heated uniformly, its area expands $$3c$$  times as fast as its radius, then the value of $$c$$ when the radius is $$6$$ units, is :

Solution :
Let $$A\,sq.$$  units in the area measure when the radius is $$r$$ units. their $$A = \pi {r^2}$$
Differentiate both side w.r.t $$'t'\,\frac{{dA}}{{dt}} = 2\pi r\frac{{dr}}{{dt}}......\left( {\text{i}} \right)$$
We have, $$\frac{{dA}}{{dt}} = 3c\frac{{dr}}{{dt}}$$
From equation $$\left( {\text{i}} \right),$$ we get $$3c.\frac{{dr}}{{dt}} = 2\pi r.\frac{{dr}}{{dt}} \Rightarrow 3c = 2\pi r$$
Now, $$c = \frac{2}{3}\pi \left( 6 \right) = 4\pi {\text{ when }}r = 6$$