When blue light of wavelength 470 falls on a single slit, the first dark bands on either side of center are separated by 45.0. determine the width of the slit

For Fraunhofer diffraction at a single slit would be represented by: 

a sinθ = mλ 

It should be noted that the angle needs be halved because we are only concerned with the angle between m=1 and m=0 and they gave you the angle between m=1 to the right of the center and m=1 on the left of the center. We calculate as follows:

a sin(45/2)=(1)(470)
a = 1228 nm 

Hope this answers the question. Have a nice day.