$4.$ Suppose we have a rocket combustor that has hot products produced at the following conditions:

T $=$ temperature $=4600$ R

P $=$ pressure $=75$ atmospheres

$=$ ratio of specific heats $=1\mathrm{.}25$

MW $=$ average molecular weight $=27$

$($a$)$ Design a nozzle that will produce $75,000$ pounds of thrust with an ambient pressure of one atmosphere. In particular, determine the following quantities: mass flow rate, exit pressure, exit or exhaust velocity, effective exhaust velocity, thrust coefficient, throat cross$-$sectional area, and exit cross$-$sectional area.

$($b$)$ Design a nozzle that produces $100,000$ pounds of thrust with an ambient pressure at vacuum conditions. Limit the nozzle exit cross$-$sectional area to no more than thirty times the throat cross$-$sectional area. Determine the same quantities as described in Part $($a$).$