find the rates of change

Question

A spherical balloon is inflated with gas at a rate of 900 cubic centimeters per minute.

(a)Find the rates of change of the radius when r = 60 centimeters and r = 95 centimeters.

r = 60 cm/min

r = 95 cm/min

Answer 1

For a sphere:

V = (4/3)π * r³

We can find the rate of change of volume with respect to the sphere's radius:

dV/dr = (4/3)π *3*r² = 4π *r² (1)

Now, we are given dV/dt, and we want to work dr/dt into (1).

Using the chain rule we can write:

dV/dt = (dV/dr)*(dr/dt) (2)

Now we can substitute dV/dr from (1) into (2), and also substitute given value for dV/dt:

900 cm³/min = (4π * r²) (dr/dt)

Solving for dr/dt:

dr/dt = 900/(4π *r²) (in units of cm/min)

At r = 60cm: dr/dt = 900/(4π *(60²)) = 0.01989 cm/min

At r = 95cm: dr/dt = 900/(4π *(95²)) = 0.00794 cm/min