For the first-order gaseous reaction $A \rightarrow 2B + C$ at $373 \, K$,starting with pure $A$,the total pressure of the system after $10 \, min$ is $176 \, mm \, Hg$ and after a long time,it is $270 \, mm \, Hg$. Based on this information,calculate: $(1)$ Initial pressure of $A$,$(2)$ Pressure of $A$ after $10 \, min$,and $(3)$ Rate constant.

• A
  $90 \, mm, 47 \, mm, 6.496 \times 10^{-2} \, min^{-1}$
• B
  $80 \, mm, 43 \, mm, 6.496 \times 10^{-5} \, min^{-1}$
• C
  $75 \, mm, 41 \, mm, 6.496 \times 10^{-2} \, min^{-1}$
• D
  $90 \, mm, 43 \, mm, 6.496 \times 10^{-7} \, min^{-1}$