(a) [tex]1.05^{\circ}[/tex]
The equation that gives the location of the minima in a diffraction pattern from a single slit is:
[tex]a sin \theta = n\lambda[/tex] (1)
where
a is the width of the slit
[tex]\theta[/tex] is the angle of the nth-minimum
[tex]\lambda[/tex] is the wavelength of the wave
Here we want to know the location of the first minimum, so n=1.
In order to find the wavelength of the sound wave, we must divide the speed of the wave by the frequency:
[tex]\lambda=\frac{v}{f}=\frac{346 m/s}{2.0\cdot 10^4 Hz}=0.0173 m[/tex]
And the slit width is
a = 0.94 m
So we can now re-arrange the eq.(1) to find the angle of the first minimum:
[tex]\theta = sin^{-1} (\frac{n\lambda}{a})=sin^{-1}(\frac{(1)(0.0173 m)}{0.94 m})=1.05^{\circ}[/tex]
(b) [tex]3.1\cdot 10^{-5} m[/tex]
This time, we have
[tex]\lambda=569 nm=5.69\cdot 10^{-7} m[/tex] is the wavelength of the yellow light
[tex]\theta=1.05^{\circ}[/tex] is the angle of the first minimum
So, re-arranging the formula for a, we find how wide the doorway should be:
[tex]a=\frac{n\lambda}{sin \theta}=\frac{(1)(5.69\cdot 10^{-7} m)}{sin(1.05^{\circ})}=3.1\cdot 10^{-5} m[/tex]
