Question #d1bd0

1 Answer
Feb 7, 2018

The rate is #96pi ~~ 301.59 \ (2dp) \ cm^2 \ mi n^(-1)#

Explanation:

Let us set up the following variables:

# { (t,"time", min), (r, "radius at time "t, cm), (A, "Area of the circle at time "t, cm^2) :} #

The area of the circle is:

# A=pir^2 #

Differentiating wrt #r#:

# (dA)/(dr) = 2pir #

Using the chain rule we can write:

# (dA)/(dt) = (dA)/(dr) \ (dr)/(dt)#

We are given that the radius #r# is expanding at #4 \ cm \ mi n^(-1)#; then #(dr)/(dt)=4# giving us:

# (dA)/(dt) = 4 \ (dA)/(dr) = 8pir#

So when #r=12# we have:

# (dA)/(dt) = 8pi xx 12 #
# \ \ \ \ \ \ = 96pi#
# \ \ \ \ \ \ ~~ 301.59 \ (2dp) \ cm^2 \ mi n^(-1)#