Answer
The smallest positive value of $x$ that corresponds to a node is $~~x = 0.50~m$
Work Step by Step
We can use superposition to find the equation for the standing wave:
$y'(x,t) = (0.050)~cos~(\pi x-4\pi t) + (0.050)~cos~(\pi x + 4\pi t)$
$y'(x,t) = (2)(0.050)~cos~(\pi x)~cos~(4\pi t)$
$y'(x,t) = (0.10)~cos~(\pi x)~cos~(4\pi t)$
We can find values of $x$, where $x \geq 0$, such that $y'(x,t) = 0$ for all values of $t$:
$y'(x,t) = (0.10)~cos~(\pi x)~cos~(4\pi t) = 0$
$cos~(\pi x) = 0$
$\pi x = 0.50 \pi, 1.5 \pi, 2.5 \pi,...$
$x = 0.50, 1.5, 2.5,...$
The smallest positive value of $x$ that corresponds to a node is $~~x = 0.50~m$
