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 \