Spherical Balloon increasing radius Calculus Problem

Question

The radius of a spherical balloon is increasing at a rate of 3 centimeters per minute. How fast is the volume changing, in cubic centimeters per minute, when the radius is 10 centimeters?

Answer 1

V=(4/3)πr³

dV/dt = 4πr² (dr/dt)

dV/dt= 4 π (10)² *3

Answer 2

The radius increases at a rate of 3cm/min or

r = ro + 3cm/min t

where ro is the unknown initial radius.

The problem is asking what is dV/dt at r = 10 cm.

The volume is

V=4/3 pi r^3

V=4/3 pi (ro + 3cm/min t)^3

dV/dt = 4x3cm/min pi (ro + 3cm/min t)^2

The above rate is in terms of t so we must find t for r= 10 cm and substitute above.

Inverting the first equation:

t = (r -ro)min/(3 cm)

Substituting for r = 10 cm:

t = 10/3 min - ro/(3 cm/min)

Substituting the above result in dV/dt and expanding, ro cancels out and the volumetric rate of expansion at r=10cm becomes:

dV/dt = 3,770 cm^3/min