To solve for the particle's position at time $t = 5$, given the acceleration, initial velocity, and initial position, we can follow these steps:

1. Given Information:

   • Acceleration: $a(t) = 36t + 12$
   • Initial Position: $s(0) = 3$
   • Initial Velocity: $v(0) = 8$

2. Step 1: Integrate Acceleration to Find Velocity
   We start by integrating the acceleration function to get the velocity function.

   $v(t) = \int a(t) \, dt = \int (36t + 12) \, dt$

   This integration yields:

   $v(t) = 18t^2 + 12t + C$

   Using the initial condition $v(0) = 8$:

   $v(0) = 18(0)^2 + 12(0) + C = 8 \Rightarrow C = 8$

   Thus, the velocity function is:

   $v(t) = 18t^2 + 12t + 8$

3. Step 2: Integrate Velocity to Find Position
   Next, we integrate the velocity function to get the position function.

   $s(t) = \int v(t) \, dt = \int (18t^2 + 12t + 8) \, dt$

   Performing this integration:

   $s(t) = 6t^3 + 6t^2 + 8t + C$

   Using the initial condition $s(0) = 3$:

   $s(0) = 6(0)^3 + 6(0)^2 + 8(0) + C = 3 \Rightarrow C = 3$

   Therefore, the position function is:

   $s(t) = 6t^3 + 6t^2 + 8t + 3$

4. Step 3: Substitute $t = 5$ to Find $s(5)$

   Now we calculate $s(5)$:

   $s(5) = 6(5)^3 + 6(5)^2 + 8(5) + 3$

   Breaking it down:

   • $6(5)^3 = 6 \cdot 125 = 750$
   • $6(5)^2 = 6 \cdot 25 = 150$
   • $8 \cdot 5 = 40$
   • Summing these up: $750 + 150 + 40 + 3 = 943$

   Therefore, the position of the particle at $t = 5$ is:

   $s(5) = 943$

Would you like further details or have any questions about this solution?

Here are five related questions to explore:

1. How would the solution change if the initial velocity $v(0)$ were different?
2. What would be the particle's position function if the acceleration were $a(t) = 12t + 6$ instead?
3. How can we determine the particle's speed at $t = 5$ using the velocity function derived?
4. What if the initial position $s(0)$ was negative—how would that impact $s(5)$?
5. How does integrating acceleration give us velocity, and integrating velocity give us position?

Tip: When working with initial conditions, double-check each value after integration to ensure the constants are applied correctly.