Step 1: Use stretch ratios instead of computing volumes directly.
In continuum mechanics, incompressibility means the product of the three stretch ratios along the three perpendicular directions equals $1$:
\[ \lambda_r \cdot \lambda_\theta \cdot \lambda_z = 1 \]
where $\lambda_z$ is the stretch along the cylinder's axis and $\lambda_r$, $\lambda_\theta$ are the stretches along the radial and circumferential directions.
Step 2: Use symmetry to simplify.
Because the cylinder is stretched only along its axis, with no directional preference in the cross-section, the radial and circumferential stretches must be equal: $\lambda_r = \lambda_\theta$. Call this common value $\lambda_{lat}$.
Substituting into the incompressibility condition:
\[ \lambda_{lat}^2 \cdot \lambda_z = 1 \implies \lambda_{lat} = \frac{1}{\sqrt{\lambda_z}} \]
Step 3: Plug in the given axial stretch.
The tissue is stretched by 10%, so $\lambda_z = 1.10$.
\[ \lambda_{lat} = \frac{1}{\sqrt{1.10}} = \frac{1}{1.0488} = 0.9535 \]
Step 4: Apply this lateral stretch ratio directly to the diameter.
Since diameter scales the same way as radius, the new diameter is the old diameter times $\lambda_{lat}$:
\[ d_1 = d_0 \times \lambda_{lat} = 2 \times 0.9535 = 1.907 \text{ cm} \]
Step 5: Round and match to the options.
$1.907$ cm rounds to $1.9$ cm, which is option (D).
Final Answer:
The stretched diameter is about $1.9$ cm.
\[ \boxed{1.9 \text{ cm}} \]