Find rate of increase of balloon diameter

Question

The volume of a spherical balloon is increasing at a rate of 2 cm³/s.  What is the rate at which the diameter of the balloon is increasing when the diameter is 10 cm?

Answer 1

What we know:

 

dV/dt = 2

D = 10

 

V = (4/3)π(D/2)³

V = (4/3)π(D³ / 8)

V = (1/6)πD³

 

What we want to find out:

 

dD/dt

 

 

We can write a formula for dV/dt.

 

dV/dt = (dV/dD) * (dD/dt)

 

dV/dt = (1/2)πD² * (dD/dt)

 

2 = (1/2)(100π) * (dD/dt)

 

2 = 50π * dD/dt

 

1/25π = dD/Dt

 

 

Diameter increases at a rate of 1/25π cm/s.