According to the Huygens-Fresnel principle, when a wave with wavelength λ travels through a slit with a width d, no quiet spots will be created outside the room due to diffraction of waves when:
[tex]d=\lambda[/tex]Therefore, since the wavelength of the sound waves is 50.0cm, the widest we can leave the door open so that everyne on the other side can hear it, is:
[tex]50.0\operatorname{cm}[/tex]