Solution:

The system shows two coaxial cylinders made of materials with different thermal conductivities (\(K_1\) and \(K_2\)) and radii \(r\) and \(3r\). For axial heat flow, the conductivities act in parallel.

To find the equivalent thermal conductivity (\(K_{\text{eq}}\)) of the entire system, we use the formula for thermal conductivities in parallel, similar to how resistances in parallel are treated for electrical systems:

\[
K_{\text{eq}} = \frac{K_1 A_1 + K_2 A_2}{A_1 + A_2}
\]

Where:
- \(A_1 = \pi r^2\) is the area of the inner cylinder.
- \(A_2 = \pi (3r)^2 - \pi r^2\  = 8\pi r^2\) is the area of the outer cylinder.

Substituting into the equation:

\[
K_{\text{eq}} = \frac{K_1 (\pi r^2) + K_2 (8\pi r^2)}{\pi r^2 + 8\pi r^2}
\]

Simplifying:

\[
K_{\text{eq}} = \frac{K_1 + 8K_2}{1 + 8} = \frac{K_1 + 8K_2}{9}
\]

Thus, the equivalent thermal conductivity is:

\[
K_{\text{eq}} = \frac{K_1 + 8K_2}{9}
\]