Use differentials to estimate the amount of ice in cubic inches that covers a 3 ft cube if the ice is $\frac{1}{2}$ inch thick.
Since it is a cube, I believe the equation should be:
$ y = x^3 $
Take the derivative:
$ y' = 3x^2 $
Plug in 3:
$ y = 3*3^2 $
Then plug in 3 + .5 (1/2 inch)
$ y = 3 * (3.5)^2 $
Resulting in:
36.75 - 27 = 9.75 inches cubed
Is this correct, or did I make a mistake somewhere along the way?
EDIT
$ y = 3 * (3)^2 * \frac{1}{12} $
Resulting in: 2.25 inches cubed. (seems more reasonable).