Two pressure vessels in a plant are connected by a series of pipes and filters. The pressures in each vessel are different in this dynamic situation.

The pressures, relative to a constant reference level, in vessel $1$ and vessel $2$ are given by p $($t$)$ and pa $(1)$ respectively at time $2.$ They are govermed by the following pair of first order, coupled differential equations:

dp dt

where a$-50,$ b$-20,120,$ d$-50$

$(1)$ Find the pressures in each pressure vessel if the initial pressures in vessel T and vessel $2$ are P$1-10$ and $2-1.$

For this part of the question give the pressure in pressure vessel $1$ as a function of t Omit the $\text{'}$py $(11$ marka$)$

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$($II$)$ What is po as a function of $1?$ Omit the $\text{'}$pa $(10$ marks$)$

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$($III$)$ What is the pressure in pressure vessel $1$ alt$=0\mathrm{.}0877$ Give your answer to AT LEAST THREE PLACES OF DECIMALS. Put in this value only ie$.$ amit the p$1(2$ marks$)$

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$($IV$)$ What is the pressure in tank $7$ at $1=0\mathrm{.}0877$

Give your ariawer to AT LEAST THREE PLACES OF DECIMALS. Put in this value only ie omit the $"$phy$(2$ marks$)$

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