Step 1: Understanding the Concept:
This is a standard "related rates" problem in calculus. We are given the rate of change of the radius of a circle with respect to time and are asked to find the rate of change of its area. We relate the two quantities with the area formula and differentiate with respect to time.
Step 2: Key Formula or Approach:
1. Formula for the area of a circle: $A = \pi r^2$.
2. Differentiate both sides with respect to time $t$ using the chain rule: $\frac{dA}{dt} = \frac{d}{dr}(\pi r^2) \cdot \frac{dr}{dt} = 2\pi r \frac{dr}{dt}$.
3. Substitute the given specific values to evaluate $\frac{dA}{dt}$.
Step 3: Detailed Explanation:
Let $r$ be the radius of the circular wave and $A$ be its enclosed area at any time $t$.
We are given:
Rate of increase of radius, $\frac{dr}{dt} = 2.1 \text{ cm/sec}$.
Specific radius at the moment of interest, $r = 10 \text{ cm}$.
Value of pi to use, $\pi = \frac{22}{7}$.
The relationship between area and radius is:
\[ A = \pi r^2 \]
Differentiate with respect to time $t$:
\[ \frac{dA}{dt} = \pi \cdot 2r \cdot \frac{dr}{dt} \]
\[ \frac{dA}{dt} = 2\pi r \frac{dr}{dt} \]
Now, plug in the given values into this rate equation:
\[ \frac{dA}{dt} = 2 \cdot \left(\frac{22}{7}\right) \cdot (10) \cdot (2.1) \]
To simplify the calculation, express $2.1$ as a fraction or note its relation to $7$:
$2.1 = \frac{21}{10}$
Substitute this back:
\[ \frac{dA}{dt} = 2 \cdot \left(\frac{22}{7}\right) \cdot 10 \cdot \left(\frac{21}{10}\right) \]
Cancel out the $10$s:
\[ \frac{dA}{dt} = 2 \cdot \left(\frac{22}{7}\right) \cdot 21 \]
Cancel $21$ with $7$ ($21 \div 7 = 3$):
\[ \frac{dA}{dt} = 2 \cdot 22 \cdot 3 \]
Multiply the remaining numbers:
\[ \frac{dA}{dt} = 44 \cdot 3 = 132 \]
The units for the rate of area change are square centimeters per second.
Step 4: Final Answer:
The rate of increase is $132 \text{ cm}^2 / \text{ second}$.