-/3 points details my notes a stone is dropped into a lake, creating a circular ripple that travels outward…

Explanation:

Step1: Find the radius - time relationship

The speed of the ripple is 40 cm/s. So the radius $r$ of the circle as a function of time $t$ is $r = 40t$.

Step2: Write the area formula of a circle

The area of a circle is $A=\pi r^{2}$. Substitute $r = 40t$ into it, we get $A=\pi(40t)^{2}=1600\pi t^{2}$.

Step3: Differentiate the area with respect to time

Using the power - rule for differentiation, if $A = 1600\pi t^{2}$, then $\frac{dA}{dt}=3200\pi t$.

Step4: Calculate the rate of change of area for different times

(a) When $t = 2$ s

Substitute $t = 2$ into $\frac{dA}{dt}=3200\pi t$. Then $\frac{dA}{dt}=3200\pi\times2 = 6400\pi$ $cm^{2}/s$.

(b) When $t = 5$ s

Substitute $t = 5$ into $\frac{dA}{dt}=3200\pi t$. Then $\frac{dA}{dt}=3200\pi\times5=16000\pi$ $cm^{2}/s$.

(c) When $t = 7$ s

Substitute $t = 7$ into $\frac{dA}{dt}=3200\pi t$. Then $\frac{dA}{dt}=3200\pi\times7 = 22400\pi$ $cm^{2}/s$.

Answer:

(a) $6400\pi$ $cm^{2}/s$ (b) $16000\pi$ $cm^{2}/s$ (c) $22400\pi$ $cm^{2}/s$