#color(purple)("Declaring the variables")#
Let a small change in radius be #deltar#
Let a small change in area be #deltaa#
Let a small change in time be #deltat#
Given that #("change in radius")/("change in time") ->(deltar)/(deltat) = 0.01/1color(white)(.) (cm)/s" "..Eqn(1)#
We are required to determine
#("change in area")/("change in time") ->(deltaa)/(deltat)#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(purple)("The logic")#
Treating this the same way you would fractions
We need to 'convert' #color(red)((delta r)/("deltat))# into #color(green)((deltaa)/(deltat))#
So if we introduce # color(blue)((deltaa)/(deltar)) # we can do this:
# color(red)((delta r)/(deltat)) color(blue)(xx(deltaa)/(deltar)) color(white)("dddd")-> color(white)("dddd") ubrace(color(red)((cancel(delta r))/(deltat))) color(blue)(xxubrace((deltaa)/(cancel(deltar)))) = color(green)((deltaa)/(deltat) " "....Eqn(2)#
#color(white)("dddddddddddddddddd") ("cm")/s xx ("cm)^2/(cm)=(cm)^2/s#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(purple)("The calculations")#
Using #Eqn(1)# substitute for #(deltar)/(deltat) #
#color(green)((deltaa)/(deltar)=(delta r)/(deltat)xx(deltaa)/(deltar)color(white)("dddd")->color(white)("dddd") (deltaa)/(deltar)=0.01/1xx(deltaa)/(deltar))#
We now need to determine #(deltaa)/(deltar)#
Known that #a=pir^2 => (deltaa)/(deltar)=2pir# so we now have:
# color(green)((deltaa)/(deltar)=0.01/1xx(deltaa)/(deltar) color(white)("ddd")->color(white)("ddd") (deltaa)/(deltar)=0.01/1xx2pir)#
But at this time #r=12cm#
#color(green)(color(white)("ddddddddddddddddd")->color(white)("ddd") (deltaa)/(deltar)=0.01/1xx2(12)pi)#
#color(green)(color(white)("ddddddddddddddddd")->color(white)("ddd") (deltaa)/(deltar)=0.24picolor(red)((cm)^2/s))#
#color(blue)("The rate of change in area at 12 cm radius is: "0.24pi (cm)^2/s)#
