Establish the relationship

Since the material is the same, weight is directly proportional to the volume of the rubber. The volume of a sphere is $V = \frac{4}{3}\pi r^3$.

$W = k \times V = k \times \frac{4}{3} \pi (\frac{D}{2})^3$

Calculate proportionality constant

For the solid sphere, $D = 6$ cm and weight $W = 6$ kg. So, $6 = k \times \frac{4}{3} \pi (3^3) = k \times 36\pi$. Thus, $k = \frac{6}{36\pi} = \frac{1}{6\pi}$.

$k = \frac{1}{6\pi}$

Determine hollow sphere volume

The volume of a hollow sphere is the difference between the outer volume and the inner volume: $V_{hollow} = \frac{4}{3}\pi (R_{outer}^3 - R_{inner}^3)$. Using diameters: $V_{hollow} = \frac{4}{3}\pi (9^3 - 6^3) = \frac{4}{3}\pi (729 - 216) = \frac{4}{3}\pi (513) = 684\pi$.

$V_{hollow} = 684\pi$

Calculate final weight

Multiply the volume by the constant $k$: $\text{Weight} = k \times V_{hollow} = \frac{1}{6\pi} \times 684\pi = 114$ kg.

$\text{Weight} = 114 \text{ kg}$