(c) Determine the total distance the weight travels from time t=0 to time t=pi Be sure to include units. (Hint: the interval [0,pi ] represents one half of a period for the function v(t)=6cos(t).)

To determine the total distance the weight travels from time to time , we need to find the area under the velocity-time graph over this interval.The velocity function is given as . The interval represents one half of a period for the function.To find the total distance need to calculate the definite integral of the velocity function over the given interval: Evaluating this integral, we get: However, since the weight oscillates back and forth, the total distance traveled is twice the absolute value of the integral over one period. Therefore, the total distance traveled over the interval is: So, the total distance the weight travels from time to time is 12 units.