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Chain rule:

To differentiate y = f(g(x)), let u = g(x). Then y = f(u) and

\[\dfrac{{dy}}{{dx}}{\text{ }} = {\text{ }}\dfrac{{dy}}{{du}}{\text{ }} \times {\text{ }}\dfrac{{du}}{{dx}}\]

Let the radius of the ring be r

So, according to question

Radius, r = 15 cm

If the radius is increasing at a constant rate of, \[\dfrac{{dr}}{{dt}} = 4cm/\sec \]

Let area be \[A = \pi {r^2}\]…………………………(1)

Differentiating the equation (1) with respect to, \[\dfrac{{dA}}{{dt}} = 2\pi r \times \dfrac{{dr}}{{dt}}\]

So, Rate of change of disturbed area = \[\dfrac{{dA}}{{dt}} = 2\pi r \times \dfrac{{dr}}{{dt}}\]

\[

\Rightarrow \dfrac{{dA}}{{dt}} = 2\pi \left( {15} \right) \times \left( 4 \right) \\

\Rightarrow \dfrac{{dA}}{{dt}} = 376.99c{m^2}/\sec \\

\]

The rate of change of disturbed area is \[376.99c{m^2}/\sec \]at the instant when the radius of the wave ring is 15 cm.