The problem presented describes a particle moving along a straight line with the position-time relationship $x(t) = -\frac{1}{2}t^2 + 23t - 204$, where time is in seconds and position is in meters. The following list of questions pertains to this motion, and I will address each one step by step.

3.1 Initial Position

The initial position is determined by evaluating $x(t)$ at $t = 0$. $x(0) = -\frac{1}{2}(0)^2 + 23(0) - 204 = -204 \text{ meters}$ Thus, the initial position is $-204$ meters.

3.2 When Does the Particle Pass the Origin?

To find when the particle passes the origin, set $x(t) = 0$ and solve for $t$. $0 = -\frac{1}{2}t^2 + 23t - 204$ Multiply the equation by 2 to simplify: $0 = -t^2 + 46t - 408$ Use the quadratic formula: $t = \frac{-46 \pm \sqrt{46^2 - 4(-1)(-408)}}{2(-1)} = \frac{-46 \pm \sqrt{2116 - 1632}}{-2} = \frac{-46 \pm \sqrt{484}}{-2} = \frac{-46 \pm 22}{-2}$ The two solutions are: $t = \frac{-46 + 22}{-2} = 12 \quad \text{and} \quad t = \frac{-46 - 22}{-2} = 34$ Therefore, the particle passes the origin at $t = 12$ seconds and $t = 34$ seconds.

3.3 When Does the Particle Exit?

By "exit," I assume the problem refers to the second time the particle crosses the origin, which is at $t = 34$ seconds, as calculated above.

3.4 When Does the Particle Change Direction?

The particle changes direction when its velocity equals zero. Velocity is the derivative of position: $v(t) = \frac{dx}{dt} = -t + 23$ Set $v(t) = 0$ to find when the direction changes: $0 = -t + 23 \quad \Rightarrow \quad t = 23 \text{ seconds}$ Thus, the particle changes direction at $t = 23$ seconds.

3.5 Maximum Positive Displacement

The maximum displacement occurs at $t = 23$ seconds (the turning point). Substituting $t = 23$ into the position equation: $x(23) = -\frac{1}{2}(23)^2 + 23(23) - 204 = -\frac{1}{2}(529) + 529 - 204 = -264.5 + 529 - 204 = 60.5 \text{ meters}$ Thus, the maximum positive displacement is 60.5 meters.

Would you like to proceed with the rest of the questions, or do you have any specific queries about this so far?