### Home > CALC3RD > Chapter Ch7 > Lesson 7.4.4 > Problem7-218

7-218.

In a hotly contested tug-of-war, the center of the rope moves back and forth according to the equation $x(t)=\frac{3}{2}-\cos(t)-\frac{1}{2}\cos(2t)$, where $x$ is measured in feet.

1. Calculate the displacement of the center of the rope over the interval $0 ≤ t ≤ π$.

$\text{displacement over time} = \text{position at end} – \text{position at start}$

2. Calculate the total distance traveled by the center of the rope over the interval $0 ≤ t ≤ π$.

If $x(t)$ was guaranteed to be strictly positive throughout the interval, then the total distance would equal the displacement.
But you were not given this guarantee, so you will have to investigate whether there is a direction change, or not.

Is there anywhere in which $x^\prime(t) < 0$?