A spherical balloon?

Question

is inflated at a rate of 14cm/s
(a) Express the radius r of the balloon as a function of the time t (in seconds). r(t)=t/14cm/s

(b) If V is the volume of the balloon as a function of the radius, find V r.

4/3pi(t/14cm/s)

not sure if that is right? do I divide t by the inflation rate?

Answer 1

We know the volume of the spherical balloon is:
V = 4(pi)r^3 /3
Normally, the inflation rate should be v/s, like cm^3/s. Please check to see if the given data is 14cm^3/s. The correct form should be: 4(pi)r^3 /3 = V = 14t
To prove this, I wonder if you have learned calculus. the inflation rate can also be written as dV/dt (check the unit to see it is correct).
inflation rate = dV/dt = 4(pi)r^2 * dr/dt
So: dr/dt = (dV/dt) / 4(pi)r^2
= 7 / [2(pi)r^2]
Or: 2(pi)r^2*dr = 7dt
Integrate both sides, considering r = 0 when t = 0:
4(pi)r^3 /3 = V = 14t, as desired. Therefore:
r(t) = cube-root of [21t /2*pi]