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To find the position \( s(5) \) of the particle at time \( t = 5 \), we need to solve this step-by-step using the provided information about acceleration, initial velocity, and initial position.

### Given:
1. **Acceleration**: \( a(t) = 36t + 12 \)
2. **Initial position**: \( s(0) = 3 \)
3. **Initial velocity**: \( v(0) = 8 \)

### Steps:
1. **Integrate the acceleration to find the velocity function** \( v(t) \).
2. **Integrate the velocity function to find the position function** \( s(t) \).
3. **Apply initial conditions** to determine any constants of integration.
4. **Substitute \( t = 5 \) into \( s(t) \)** to find \( s(5) \).

---

#### Step 1: Integrate \( a(t) \) to find \( v(t) \)

\[
v(t) = \int a(t) \, dt = \int (36t + 12) \, dt
\]
\[
v(t) = 18t^2 + 12t + C_1
\]

Using the initial condition \( v(0) = 8 \):
\[
v(0) = 18(0)^2 + 12(0) + C_1 = 8 \Rightarrow C_1 = 8
\]
Thus, the velocity function is:
\[
v(t) = 18t^2 + 12t + 8
\]

---

#### Step 2: Integrate \( v(t) \) to find \( s(t) \)

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

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?

Learn how to find the position of a particle with a given acceleration function a(t) = 36t + 12, initial position s(0) = 3, and initial velocity v(0) = 8 at t = 5. This problem involves calculus concepts such as integration and initial conditions, ideal for college-level students studying kinematics.

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