To solve this problem it is necessary to apply the concepts related to the principle of superposition and the equations of destructive and constructive interference.
Constructive interference can be defined as
[tex]dSin\theta = m\lambda[/tex]
Where
m= Any integer which represent the number of repetition of spectrum
[tex]\lambda[/tex]= Wavelength
d = Distance between the slits.
[tex]\theta[/tex]= Angle between the difraccion paterns and the source of light
Re-arrange to find the distance between the slits we have,
[tex]d = \frac{m\lambda}{sin\theta }[/tex]
[tex]d = \frac{2*536*10^{-9}}{sin(24)}[/tex]
[tex]d = 2.63*10^{-6}m[/tex]
Therefore the number of lines per millimeter would be given as
[tex]\frac{1}{d} = \frac{1}{2.63*10^{-6} }[/tex]
[tex]\frac{1}{d} = 379418.5\frac{lines}{m}(\frac{10^{-3}m}{1 mm})[/tex]
[tex]\frac{1}{d} = 379.4 lines/mm[/tex]
Therefore the number of the lines from the grating to the center of the diffraction pattern are 380lines per mm
