The radius of a spherical balloon being filled with air expands at 4 cm^3 per minute. Assuming the balloon fills in spherical shape, how fast is the radius of the spherical balloon increasing in cm per minute after 2.25 minutes?

Answer: dr/dt = 0.042 cm/minute

Step-by-step explanation:

Given;

dV/dt = 4cm^3/minute

t = 2.25minutes

Volume of a sphere is given as;

V = (4/3) πr^3

Change in Volume ∆V can be derived by differentiating the function.

dV/dt = 4πr^2. dr/dt

dV/dt = 4πr^2dr/dt ... 1

dV/dt is given as 4 cm^3/min

radius after 2.25 minutes can be gotten from the the volume.

Volume after 2.25mins = 4*2.25 = 9cm^3

9cm^3 = V = 4/3πr^3

r^3 = 27/4π

r = (27/4π) ^1/3

From equation 1.

dr/dt = (dV/dt) / 4πr^2 = 4 / (4πr^2) = 1 / (πr^2)

dr/dt = 1 / (π (27/4π) ^2/3)

dr/dt = 0.042cm/minute.