Question:

Step 1: Identify the relationship between area and radius. The area of a circle is given by the formula: $$A = \pi r^2$$ Step 2: Differentiate the area with respect to time. To find the rate of change of area with respect to time, we differentiate both sides of the equation with respect to $t$: $$\frac{dA}{dt} = \frac{d}{dt}(\pi r^2) = 2\pi r \frac{dr}{dt}$$ This uses the chain rule, since $r$ is a function of time $t$. Step 3: Extract the given information. From the problem statement, we identify: - The speed at which the wave moves (rate of increase of radius): $\dfrac{dr}{dt} = 5$ cm/sec - The radius at the instant in question: $r = 8$ cm Step 4: Substitute the values into the differentiated equation. Now we substitute $r = 8$ cm and $\dfrac{dr}{dt} = 5$ cm/sec into the equation from Step 2: $$\frac{dA}{dt} = 2\pi (8)(5) = 80\pi \text{ cm}^2\text{/sec}$$ Step 5: State the final answer. The rate of increase of the enclosed area when the radius is 8 cm is $80\pi$ cm²/sec. This corresponds to **Option 4: (d) $80\pi$ cm²/sec**.