Water leaking onto a floor forms a circular pool. The area of the pool increases at a rate of 25π cm²/min. How fast is the radius of the pool increasing when the radius is 6 cm?

1 Answer
Oct 23, 2016

#dr/dt = (25)/(12)#cm/min

Explanation:

so from the question, we know that #dA/dt=25pi#, this means that the area of the circular puddle is increasing constantly at this rate.

so in order to find how fast the radius is increasing, we must first determine a relationship between the two values. So, for a circle that is the Area formula #Area = pi*r^2#.

The next part involves related rates and the chain rule.
We know that #dr/dt =dA/dt * dr/dA#
(the rate of radius change expressed as two other rates).
So,
#A=pi*r^2 #

#dA/dr = 2pi*r#

#dr/dA = 1/(2pi*r)#

using chain rule now.
#dr/dt = dA/dt *dr/dA#

#dr/dt =25pi * 1/(2pi*r)#

Sub in 6cm for radius
#dr/dt = (25pi)/(2*6pi)#
#dr/dt = (25)/(12)#cm/min