A rigid body rotates about a fixed axis with variable angular velocity ($\propto-\beta \mathrm{t}$) at time t, where $\propto$ and $\beta$ are constants. The angle through which it rotates before it comes to rest is

The angular velocity is $\omega = \alpha - \beta t$. The body reaches rest when $\omega = 0$, giving $\alpha - \beta t_f = 0$ and $t_f = \frac{\alpha}{\beta}$.

The total rotation angle is the integral of angular velocity over time:

$\theta = \int_0^{\alpha/\beta} (\alpha - \beta t) \, dt = \left[ \alpha t - \frac{\beta t^2}{2} \right]_0^{\alpha/\beta}$

Evaluating at the limits:

$\theta = \left( \alpha \cdot \frac{\alpha}{\beta} - \frac{\beta}{2} \cdot \frac{\alpha^2}{\beta^2} \right) - 0 = \frac{\alpha^2}{\beta} - \frac{\alpha^2}{2\beta} = \frac{\alpha^2}{2\beta}$

Final answer: $\boxed{\text{C}}$