As Khalid inflates a spherical balloon, Kareem wonders about its different rates. He knows that the rate at which Khalid blows is equal to the rate at which the volume changes $( \frac { d V } { d t } )$. As the balloon inflates, other aspects are changing as well, such as the radius and the surface area.

1. If $\frac { d V } { d t }= 10\frac {\text{cm} ^ { 3 } } {\text{sec} }$, find the rate of change of the radius, $\frac { d r } { d t }$, when $r = 3$ cm.

   Step 1(a):

   Using just geometry (no Calculus), find the equation for a generic spherical balloon.
   $V=\frac{4}{3}\pi r^{3}$

   Step 2(a):

   Substitute the given information and solve.

2. If $\frac { d V } { d t }= 12\frac {\text{cm} ^ { 3 } } {\text{sec} }$, find the rate of change of the surface area, $\frac { d A } { d t }$, when $r = 5$ cm.

   Hint (b):

   Find $\frac{dr}{dt}$ , you will need that information later.

   Step 1(b):

   Using just geometry, write the equation for the Surface Area of a generic spherical balloon: $A = 4πr^²$

3. Describe what happens to the balloon when $\frac { d V } { d t }$ is negative.

   Hint (c):

   What does a negative rate represent?