Answer
$x = 0.50~m$
Work Step by Step
We can use superposition to find the equation for the standing wave:
$y'(x,t) = (6.0~cm)~cos~\frac{\pi}{2}[(2.00 ~m^{-1})x+(8.00~s^{-1})~t] + (6.0~cm)~cos~\frac{\pi}{2}[(2.00 ~m^{-1})x-(8.00~s^{-1})~t]$
$y'(x,t) = (12~cm)~cos~\frac{\pi}{2}[(2.00 ~m^{-1})x]~\cdot cos~\frac{\pi}{2}[(8.00~s^{-1})~t]$
We can find the values of $x$ such that $y'(x,t) = 0$ for all values of $t$:
$cos~\frac{\pi}{2}[(2.00 ~m^{-1})x] = 0$
$cos~[(\pi m^{-1})x] = 0$
$(\pi m^{-1}) x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, ...$
$x = \frac{1}{2} m, \frac{3}{2} m, \frac{5}{2} m, ...$
$x = 0.50 ~m, 1.5 ~m, 2.5 ~m, ...$
The location of the node having the smallest value of $x$ is $~~x = 0.50~m$
