At a given instant, the radii of two concentric circles aree 8cm and 12cm. The radius of the outer circle is increasing at a rate of $\displaystyle 1cms^{-1} $

and the radius of the inner circle is increasing at a rate of $\displaystyle 2cms^{-1} $. Find the rate of change of the area enclosed by the two circles.

$\displaystyle r = 12, \frac{dr}{dt} = 1 $

$\displaystyle r = 8 \frac{dr}{dt} = 2 $

$\displaystyle \frac{dA}{dt} = \frac{dA}{dr} \times \frac{dr}{dt} $

$\displaystyle A = \pi r^{2} $

$\displaystyle \frac{dA}{dr} = 2\pi r $

is this all correct? So I would just add the two da/dt for each circle?

I get $\displaystyle 56\pi $

the correct answer is $\displaystyle 8\pi $

Can someone tell me where I am going wrong? Thanks.