Q.6) If the radius of the circle is increasing at the rate of $$0.5 \mathrm{~cm} / \mathrm{s}$$, then the rate of increase of its circumference is $$\_\_\_\_$$。

Solution:

Rate of increase of circumference

Step 1: Define variables and formulas
Let $$r$$ be the radius of the circle and $$C$$ be its circumference. The formula for the circumference of a circle is $$C=2 \pi r$$. The given rate of increase of the radius is $$\frac{d r}{d t}=0.5 \mathrm{~cm} / \mathrm{s}$$.

Step 2:
Differentiate the circumference formula with respect to time $$t$$ to find the relationship between the rates of change:
$$
\begin{gathered}
\frac{d C}{d t}=\frac{d}{d t}(2 \pi r) \\
\frac{d C}{d t}=2 \pi \frac{d r}{d t}
\end{gathered}
$$

Step 3: Substitute the given rate
Substitute the value of $$\frac{d r}{d t}$$ into the differentiated equation:
$$
\begin{gathered}
\frac{d C}{d t}=2 \pi(0.5 \mathrm{~cm} / \mathrm{s}) \\
\frac{d C}{d t}=1 \pi \mathrm{~cm} / \mathrm{s}
\end{gathered}
$$

Answer:
The rate of increase of its circumference is $$\pi \mathrm{cm} / \mathrm{s}$$.