$4.(12$ pts$)$ A computer system consists of three critical components: a mother board, a power supply, and a hard disk. The failure time of each component follows an exponential distribution with MTBF as follows:

Mother board $500,000$ hrs

Power supply $250,000$ hrs

Hard drive $250,000$ hrs

The U$.$S$.$ Government is purchasing $10,000$ such computers from Dell. Each computer will be used $1,460$ hours a year on average, and failures occurring during the first year of operations are automatically covered by Dell free of charge.

Dell is considering offering only one of the following two extended warranty plans to the U$.$S$.$ Government.

Plan $1$ charges $$90$ per machine for a $3-$year extended warranty, which will cover the second and third year failures.

Plan $2$ charges $$100$ per machine for a $4-$year extended warranty, which will cover the second, third, and fourth year failures.

It will cost Dell on average $$225$ per failure to repair.

Strictly from a profit perspective, which extended warranty plan $(1$ or $2$ above$)$ should Dell offer to the U$.$S$.$ Government? Justify your answer.