We are given that the circumference $C$ of the rubber band is increasing at a constant rate of $\frac{dC}{dt} = 5 \, \text{cm/sec}$. We need to find the rate at which the radius $r$ of the rubber band is changing, $\frac{dr}{dt}$, when the radius is 3 cm.

Step 1: Relate the circumference and the radius.

From the formula for the circumference of a circle: $C = 2\pi r$ Differentiate both sides with respect to time $t$: $\frac{dC}{dt} = 2\pi \frac{dr}{dt}$ We are given that $\frac{dC}{dt} = 5 \, \text{cm/sec}$, and we want to find $\frac{dr}{dt}$ when $r = 3 \, \text{cm}$.

Step 2: Solve for $\frac{dr}{dt}$.

Substitute $\frac{dC}{dt} = 5 \, \text{cm/sec}$ into the equation: $5 = 2\pi \frac{dr}{dt}$ Solve for $\frac{dr}{dt}$: $\frac{dr}{dt} = \frac{5}{2\pi}$

Step 3: Calculate the rate.

Now, simplify the expression: $\frac{dr}{dt} \approx \frac{5}{6.2832} \approx 0.796 \, \text{cm/sec}$

Thus, the radius is increasing at a rate of approximately $0.796 \, \text{cm/sec}$ when the radius is 3 cm.

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. How would the rate change if the radius was 5 cm instead of 3 cm?
2. What is the significance of the constant rate of circumference increase?
3. How does the relationship between radius and circumference affect the rate of change?
4. What if the circumference increased at a non-constant rate? How would the approach differ?
5. Can we generalize this problem to non-circular shapes? How would the rate of change behave?

Tip: When differentiating, always make sure each variable is expressed in terms of the correct independent variable, especially if multiple variables change with time.