The radius of a circle is increasing at the rate of 0.4 cm/ s. The rate of increasing of its circumference is

The correct option is 0.8 π cm/s.

Let us denote the radius of the circle as $r$ and its circumference as $C$.

We are given that the radius of the circle is increasing at a rate of 0.4 cm/s. Mathematically, this rate of change of radius with respect to time $t$ is written as:
$\frac{dr}{dt}=0.4\text{ cm/s}$

The formula for the circumference $C$ of a circle in terms of its radius $r$ is:
$C$ = 2 π $r$

To find the rate at which the circumference is increasing, we need to find the derivative of the circumference $C$ with respect to time $t$, which is $\frac{dC}{dt}$. Differentiating both sides of the circumference formula with respect to $t$ using the chain rule, we get:
$\frac{dC}{dt}=\frac{d}{dt}(2\pi r)$
$\frac{dC}{dt}=2\pi \frac{dr}{dt}$

Now, we substitute the given value of $\frac{dr}{dt}=0.4\text{ cm/s}$ into the equation:
$\frac{dC}{dt}=2\pi \times 0.4$
$\frac{dC}{dt}=0.8\pi \text{ cm/s}$

Therefore, the rate of increase of the circumference of the circle is 0.8 π cm/s.