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We are not provided with information about how the PC connects to the network, so we can assume it receives an IP address via DHCP.
When the user from the PC me.cs.bsu.edu first tries to access www.googe.com via its browser, the computer first interrogates the local DNS server residing in the same network with it - 10.4.82.251 - requesting the IP address of the domain www.google.com.
The local DNS server looks into the records to see whether it has the required IP address. If it has it, then it transmits the address to the me.cs.bsu.edu computer....