A pebble is dropped into a still pond, causing ripples in the form of concentric circles. The radius r of the outer circle is increasing at a rate of 25cm/s. When the radius is 100cm, at what rate is the total area of the disturbed water increasing?

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Answer 1 · 2017-08-14T13:21:39

#5000pi# #cm^2#/#s#

Variables

#r# = the radius of the circle (in #cm#)

#A# = the area of the circle (in #cm^2#)

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

Rates of change
#(dr)/dt = 25# #cm#/#s#

Find #(dA)/dt# when #r = 100# #cm#

Equation relating the variables ( The variables other than #t#)

#A = pir^2#

To finish

Differentiate with respect to #t#. Then plug in what we know and solve for what we want.

#d/dt(A) = d/dt(pir^2)#

#(dA)/dt = 2pir (dr)/dt# #" "# (Using the chain rule on the right.)

#(dA)/dt = 2pi(100) (25)# #cm^2#/#s#

# = 5000pi# #cm^2#/#s#