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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.

A particle is moving with acceleration a(t) = 36t + 12. Its position at time t = 0 is s(0) = 3, and its velocity at time t = 0 is v(0) = 8. What is its position at time t = 5?

Find the position of a particle at time t = 5 given its acceleration a(t) = 36t + 12, initial velocity v(0) = 8, and initial position s(0) = 3. Learn how to integrate acceleration to find velocity and position using initial conditions.

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