Relate the angle at the center to the angle at the circumference.

The angle subtended by an arc at the center is twice the angle subtended by the same arc at the circumference. For arc PR, the angle at the center is $\angle PSR$ and the angle at the circumference is $\angle PQR$.

$$\angle PSR = 2 \times \angle PQR$$

Substitute the given expression for $\angle PQR$:

$$\angle PSR = 2(2x - 40^\circ) = 4x - 80^\circ$$

Since PT is a diameter and S is the center, P, S, and T are collinear. Therefore, $\angle PST$ is a straight angle, which means $\angle PST = 180^\circ$. The angles $\angle PSR$ and $\angle RST$ are adjacent angles on this straight line.

$$\angle PSR + \angle RST = 180^\circ$$

Substitute the expressions for $\angle PSR$ and $\angle RST$ into the equation from Step 2:

$$(4x - 80^\circ) + (x - 8^\circ) = 180^\circ$$

Combine like terms:

$$5x - 88^\circ = 180^\circ$$

Add $88^\circ$ to both sides:

$$5x = 180^\circ + 88^\circ$$ $$5x = 268^\circ$$

Divide by 5:

$$x = \frac{268}{5}$$ $$x = \boxed{\text{53.6}}$$