$13.$ Researchers in microelectronics have found that a slight scratch on the surface of gallium arsenide causes a zinc dopant to diffuse into the arsenide. Apparently, this occurs because the scratch increases crystal defects and hence the local diffusion coefficient. When these devices are later baked at $850\backslash$deg C$,$ the small pulse may spread, for its diffusion coefficient at this high temperature is about $1011$ cm$2/$sec$.$ If it spreads enough to increase the zinc concentration to $10$ percent of the maximum at $4104$ cm away from the scratch, the device is ruined. How long can we bake the device? $($S$.$ Balloge$)$ Answer: $30$ min.