To find the position function for the particle, you need to integrate the acceleration function with respect to time. Let's go through the process step by step:
1. Start with the given acceleration function: a(t) = cos(t).
2. To find the velocity function, integrate the acceleration function with respect to time. The integral of cos(t) with respect to t is sin(t). So, the velocity function v(t) is given by v(t) = sin(t) + C, where C is the constant of integration.
3. To find the position function, integrate the velocity function with respect to time. The integral of sin(t) with respect to t is -cos(t). So, the position function x(t) is given by x(t) = -cos(t) + D, where D is the constant of integration.
4. To determine the constant of integration, use the initial condition given in the problem. At t = 0, the position is x = 3. Substitute these values into the position function: 3 = -cos(0) + D. Since cos(0) = 1, we have 3 = -1 + D. Solving for D, we get D = 4.
5. Finally, substitute the value of D back into the position function. Therefore, the position function for the particle is x(t) = -cos(t) + 4.