$3\mathrm{.}14.$ To reduce gas storage costs, two companies, A and $B,$ have built a common storage tank

in the shape of a horizontal right circular cylinder $0\mathrm{.}3m$ in diameter and $30m$ long $($see

Figure P$3\mathrm{.}14).$ To decide how much gas each company uses between refills, a thin piston

was placed in the tank.

The piston moves freely, that is$,$ there is essentially no friction present, and the pressure

is the same on both sides. Thus, as company A uses gas, the piston moves left and as

company B uses gas, the piston moves right.

When the gas company refills the tank, it must decide how much gas has been used by

each company. It can easily measure the position of the piston and can, if necessary, install

other instrumentation such as thermometers or pressure gauges in either or both ends of

the tank.

Figure P$3\mathrm{.}14$

Assume that $(1)$ the gas is ideal; $(2)$ the piston is adiabatic; $(3)$ the walls are well insulated

and have low heat capacity; and $(4)$ at the start of each month after filling the tank, the

gas company positions the piston in the center of the tank, equalizes the temperature on

both ends, and carefully meters the total amount of gas added.