A wood turner shapes a solid ornament on a lathe…

Step 1. The widest radius. $x(6 - x)$ is largest at $x = 3$, where it equals $9$. So $r(3) = 2 + 1 = 3$ cm.

Step 2. The volume. $V = \pi\displaystyle\int_0^6 \left(2 + \dfrac{6x - x^2}{9}\right)^2 dx$. Expanding, $r^2 = 4 + \dfrac{4(6x - x^2)}{9} + \dfrac{(6x - x^2)^2}{81}$. Integrating each term from $0$ to $6$ gives $24 + \dfrac{4 \times 36}{9} + \dfrac{1}{81}\displaystyle\int_0^6 (6x - x^2)^2\,dx$. The last integral equals $\dfrac{6^5}{30} = 259.2$, so the third term is $3.2$. The total is $24 + 16 + 3.2 = 43.2$, and $V = 43.2\pi = \dfrac{216\pi}{5} \approx 135.72$ cm$^3$.

Step 3. The mass. $0.7\ \text{g/cm}^3 \times 135.72\ \text{cm}^3 \approx 95.00$ g.

Step 4. The cylinder. Radius $3$ cm and height $6$ cm give $\pi \times 9 \times 6 = 54\pi$ cm$^3$. The fraction left is $\dfrac{43.2\pi}{54\pi} = 0.8$.