Answer
$k = 31.4~m^{-1}$
Work Step by Step
We can find the phase shift $\phi$ when the amplitude of the resultant wave is zero:
$y' = 2y_m~cos \frac{\phi}{2} = 0$
$cos \frac{\phi}{2} = 0$
$\frac{\phi}{2} = \frac{\pi}{2}$
$\phi = \pi~rad$
A phase-shift of $\pi~rad$ is equivalent to a phase-shift of $\frac{\lambda}{2}$
We can find $\lambda$:
$\frac{\lambda}{2} = 10~cm$
$\lambda = 20~cm$
We can find $k$:
$k = \frac{2\pi}{\lambda}$
$k = \frac{2\pi}{0.20~m}$
$k = 31.4~m^{-1}$
