Answer
$1.27\;cm$
Work Step by Step
A sinusoidal wave traveling along a string in the positive direction of an x axis has the mathematical form
$y(x, t) =y_m\sin(kx-\omega t)....................(1)$
where $y_m$ is the amplitude of the wave, $k$ is the angular wave number, $\omega$ is the angular frequency.
The wavelength $\lambda$ is related to $k$ by $k=\frac{2\pi}{\lambda}$
Slope:
$\frac{dy}{dx}=ky_m\cos(kx-\omega t)$
Therefore,
$\Big(\frac{dy}{dx}\Big)_{max}=ky_m=\frac{2\pi}{\lambda}y_m\;....................(2)$
From the given figure, we obtain
$\lambda=0.4\;m$ and $\Big(\frac{dy}{dx}\Big)_{max}=0.2$
Substituting the values in Eq. $(2)$, we obtain
$\frac{2\pi}{0.4}y_m=0.2$
or, $y_m=\frac{0.2\times0.4}{2\pi}$
or, $y_m=0.0127\;m=1.27\;cm$
$\therefore$ The amplitude of the wave is $1.27\;cm$
