To solve this problem, we need to use the concept of inverse variation.

Since \( P \) varies inversely as the cube root of \( R \), we can write:

\[
P = \frac{k}{\sqrt[3]{R}}
\]

where \( k \) is a constant.

Given:
- \( P = 1 \) when \( R = 8 \)

Substitute these values into the equation to find \( k \):

\[
1 = \frac{k}{\sqrt[3]{8}}
\]

Since \( \sqrt[3]{8} = 2 \):

\[
1 = \frac{k}{2}
\]

Solving for \( k \):

\[
k = 2
\]

Now, we need to find \( R \) when \( P = 3 \). Using the equation with the constant \( k = 2 \):

\[
3 = \frac{2}{\sqrt[3]{R}}
\]

Solve for \( \sqrt[3]{R} \):

\[
\sqrt[3]{R} = \frac{2}{3}
\]

To find \( R \), cube both sides of the equation:

\[
R = \left(\frac{2}{3}\right)^3 = \frac{8}{27}
\]

So, the correct answer is C. 8/27.