An innovative engineer has invented a device to replace the hydraulic jacks found at many service stations. A movable piston with a diameter of $0\mathrm{.}15$ m is fitted into a cylinder. Cars are raised by opening a small door near the base of the cylinder, inserting a block of dry ice $($solid CO$2),$ closing and sealing the door, and vaporizing the dry ice by applying just enough heat to raise the cylinder contents to ambient temperature $(25\backslash$deg C$).$ The car is subsequently lowered by opening a valve and venting the cylinder gas.

The device is tested by raising a car a vertical distance of $1\mathrm{.}5$ m$.$ The combined mass of the piston and the car is $5500$ kg$.$ Before the piston rises, the cylinder contains $0\mathrm{.}030$ m$3$ of CO$2$ at ambient temperature and pressure $(1$ atm$).$ Neglect the volume of the dry ice.

$($a$)$ Calculate the pressure in the cylinder when the piston comes to rest at the desired elevation.

$($b$)$ How much dry ice $($kg$)$ must be placed in the cylinder? Use Van der Waals equation of state for this calculation.

$($c$)$ Outline how you would calculate the minimum piston diameter required for any elevation of the car to occur if the calculated amount of dry ice is added. $($Just give formulas and describe the procedure you would follow$$no numerical calculations are required.$)$