Question

# The radius of a circle is increasing at the rate of 0.7cm/s. What is the rate of increase of its circumference?

Hint: We can find out the rate of change by just differentiating the equation with respect to time. We know the formula for circumference. Apply differentiation to find the answer.

Given, rate of change of radius of circle with time $\dfrac{{dr}}{{dt}} = + 0.7cm/s$

Since, radius is increasing with time that's why positive sign is used for $\dfrac{{dr}}{{dt}}$

As, we know that circumference of the circle $C = 2\pi r$

Therefore, rate of change of circumference can be obtained by differentiating the above equation with respect to time

i.e, Rate of change of circumference $\dfrac{{dC}}{{dt}} = \dfrac{d}{{dt}}\left( {2\pi r} \right)$

Here, $2\pi$ can be taken outside as it is a constant independent of time.

$\Rightarrow \dfrac{{dC}}{{dt}} = 2\pi \dfrac{{dr}}{{dt}}$

Now, substitute the value for $\dfrac{{dr}}{{dt}}$ and using $\pi = \dfrac{{22}}{7}$

$\Rightarrow \dfrac{{dC}}{{dt}} = 2\pi \dfrac{{dr}}{{dt}} = 2 \times \dfrac{{22}}{7} \times 0.7 = 4.4cm/s$

Since, the above rate $\dfrac{{dC}}{{dt}}$ comes out to be positive that means, the circumference is increasing with time and rate of increase of circumference is 4.4cm/s.

Note - These types of problems can be solved by simply representing the rate of change of variable (that needs to be found) in terms of the rate of change of another variable which is given in the problem by using some relation between these two variables.