In a single$-$slit diffraction experiment, a coherent light source illuminates a slit in a barrier, and the resulting pattern is projected on a screen that is separated from the barrier by a distance which is very large as compared to the slit width and the wavelength of the light source. A light sensor is used to measure the intensity of the light at the screen, and a student makes a graph, as shown below, of the intensity as a function of the position along the screen as measured from the center of the central maximum. The intensity of the central maximum is reduced by a factor of $10$ relative to the other maxima to make the graph more easily readable. The distance from the barrier to the screen is $2\mathrm{.}2$m$,$ and the light source has a wavelength of $565$nm

Part $($a$)$

What is the ratio of the slit width to the wavelength of the incident light?

Part $($b$)$

What is the width of the slit, in micrometers?

Part $($c$)$

If the screen is very wide, how many dark bands appear to one side of the central maximum?