A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius of the outer ripple is increasing at a constant rate of 6 inches per second. When the radius is 4 feet, at what rate (in ft2/sec) is the total area A of the disturbed water changing?

dA/dt = 32π ft²/s

Step-by-step explanation:

r = the radius of the circle (in ft)

A = the area of the circle (in ft²)

t = time since the pebble hit the water (in s)

Rates of change;

dr/dt = 6 in/s = 6 x 0.0833 ft/s = 0.5 ft/s

Now, we are to find dA/dt when r = 4 ft

Area (A) = πr²

Let's Differentiate with respect to

t

Thus,

dA/dt = (d/dt) (πr²)

Using chain rule,

dA/dt = (2πr) (dr/dt)

Plugging in the relevant values to get;

dA/dt = (2 x π x 4) / 0.5 = 32π ft²/s