A spherical balloon has a leak and its radius is decreasing by 2 centimeters per minute.

Question

What is the rate of change of the volume of the balloon when the radius is 40 centimeters? (V= 4/3(pi)(r)^3

 

I know how to find the Volume of the balloon at specific radii but I don't know how to find the rate of change.

Answer 1

V = (4/3)·pi·r³

• V = Volume of the balloon
• r = radius of the balloon

 

Take the derivative of V wrt t using the chain rule:

 

dV/dt = (dV/dr)·(dr/dt) = (4/3)·pi·3r²·(dr/dt)

dV/dt = 4·pi·r²·(dr/dt)

 

Now dr/dt is given as -2 cm/min.  When r = 40 cm:

 

dV/dt = 4·pi·(40)²·(-2)

 

Can you finish it from here?