How far apart are the teeth peaks on the saw given by the equation?

Question

The teeth of a hand saw can be approximated by the function $$y=x+4+4\sin(6x), 0\leq x\leq 4\pi$$ where $y$ cm is the vertical height of the teeth at a horizontal distance $x$ cm from the end of the saw. How far apart are the successive peaks of the teeth?

Answer 1

Successive peaks are separated by $\frac {2\pi}6$ in $x$. The $x$ term in the function makes each successive peak be higher than the last by $\frac {2\pi}6$ as well. The spatial distance between successive peaks is therefore $\sqrt 2 \frac \pi 3$

Answer 2

You want the period of $\sin(6x)$ which is $ 2\pi /6 = \pi /3$