(a) The volume of the balloon after 30 seconds is given by:

\(V = 600 \times 30 = 18000 \text{ cm}^3\)

The formula for the volume of a sphere is:

\(V = \frac{4}{3} \pi r^3\)

Equating the two expressions for volume:

\(\frac{4}{3} \pi r^3 = 18000\)

Solving for \(r\):

\(r^3 = \frac{18000 \times 3}{4 \pi}\)

\(r = \sqrt[3]{\frac{54000}{4 \pi}}\)

\(r \approx 16.3 \text{ cm}\)

(b) The rate of change of volume with respect to radius is:

\(\frac{dV}{dr} = 4 \pi r^2\)

Using the chain rule, the rate of change of radius with respect to time is:

\(\frac{dr}{dt} = \frac{dr}{dV} \times \frac{dV}{dt}\)

Substituting the known values:

\(\frac{dr}{dt} = \frac{1}{4 \pi r^2} \times 600\)

At \(r = 16.3 \text{ cm}\):

\(\frac{dr}{dt} = \frac{600}{4 \pi (16.3)^2}\)

\(\frac{dr}{dt} \approx 0.181 \text{ cm per second}\)