(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 $t=0$ to time $t=\pi$, we need to find the area under the velocity-time graph over this interval.<br /><br />The velocity function is given as $v(t)=6\cos(t)$. The interval $[0,\pi]$ represents one half of a period for the function.<br /><br />To find the total distance need to calculate the definite integral of the velocity function over the given interval:<br /><br />$\int_{0}^{\pi} 6\cos(t) \, dt$<br /><br />Evaluating this integral, we get:<br /><br />$\int_{0}^{\pi} 6\cos(t) \, dt = 6\sin(t)\bigg|_{0}^{\pi} = 6\sin(\pi) - 6\sin(0) = 0 - 0 = 0$<br /><br />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 $[0,\pi]$ is:<br /><br />$2 \times 6 = 12$<br /><br />So, the total distance the weight travels from time $t=0$ to time $t=\pi$ is 12 units.