Paolo wants to surprise his girlfriend Jessica (who loves chocolate) by baking and decorating a cake for her to celebrate their anniversary. The frosting bag is in the shape of a right cone with a radius equal to its height. He squeezes the bag with constant pressure so that he applies $15 \text{ cm}^3$ of frosting per minute.

1. How much frosting will Paolo have put on the cake after $42$ seconds?

   Hint (a):

   We know that Paolo squeezes $15$ cubic centimeters of frosting every $60$ seconds.

2. How fast is the radius of the frosting bag changing at the moment the radius is $1.7$ cm?

   Hint (b):

   $\text{Since the radius equals the height, }V= \frac{1}{3}\pi r^{3}.$

   $\text{Differentiate implicitly, then solve for }\frac{dr}{dt}.$

   More Help (b):

   $\text{Is }\frac{dV}{dt}\text{ positive or negative?}$

3. Will Jessica like the cake?

   Hint (c):

   What type of cake does Jessica like?