Given the formula \(M = kr^3\), we need to find \(k\) when \(M = 3.2\) kg and \(r = 10\) cm.

Substitute the values into the formula:

\(3.2 = k \times 10^3\)

\(3.2 = 1000k\)

\(k = \frac{3.2}{1000} = 0.0032\)

Next, we find the rate at which the mass is increasing, \(\frac{dM}{dt}\).

Differentiate \(M = kr^3\) with respect to \(r\):

\(\frac{dM}{dr} = 3kr^2\)

Using the chain rule, \(\frac{dM}{dt} = \frac{dM}{dr} \times \frac{dr}{dt}\).

Given \(\frac{dr}{dt} = 0.1\) cm/day, substitute the values:

\(\frac{dM}{dt} = 3 \times 0.0032 \times 10^2 \times 0.1\)

\(\frac{dM}{dt} = 0.096 \text{ kg/day}\)