To find the rate at which the circumference of a circle is increasing, given that the radius is increasing at a rate of $0.5 \, \text{cm/s}$, we can follow these steps:

Step 1: Relate the circumference to the radius

The circumference $C$ of a circle is given by the formula: $C = 2\pi r$ where $r$ is the radius of the circle.

Step 2: Differentiate the circumference with respect to time

To find the rate of change of the circumference with respect to time, we differentiate the equation $C = 2\pi r$ with respect to time $t$: $\frac{dC}{dt} = \frac{d}{dt}(2\pi r)$ Since $2\pi$ is a constant, this simplifies to: $\frac{dC}{dt} = 2\pi \frac{dr}{dt}$

Step 3: Substitute the given rate of change of the radius

We are given that the radius is increasing at a rate of $0.5 \, \text{cm/s}$, so $\frac{dr}{dt} = 0.5 \, \text{cm/s}$.

Substituting this value into the equation for $\frac{dC}{dt}$: $\frac{dC}{dt} = 2\pi \times 0.5 \, \text{cm/s} = \pi \, \text{cm/s}$

Final Answer

The rate at which the circumference of the circle is increasing is:

$\boxed{\pi \, \text{cm/s}}$

Would you like any more details or have further questions?

Here are five related questions you could explore:

1. How does the rate of change of the area of the circle compare to the rate of change of its circumference?
2. If the rate of change of the radius were to double, what would be the new rate of change of the circumference?
3. What would be the rate of change of the diameter of the circle?
4. How does the rate of change of the radius affect the overall growth of the circle's area over time?
5. If the rate of change of the radius is a function of time, how would you find the rate of change of the circumference at a specific time?

Tip: When dealing with related rates, always ensure you differentiate the relevant formulas with respect to time, keeping in mind the quantities that change with time.