Fluid mechanics of aircraft at 10000 m 

  
*An aircraft cruises at $245\,\rm m/s$ at an altitude of $10\,000\,\rm m$ where the air density is $0.414\,\rm kg/m^3$. Each engine has an intake area of $5.50\,\rm m^2$ and air enters the engine at a relative velocity of $215\,\rm m/s$. The exhaust gas density is $0.343\,\rm kg/s$ and the jet exit area is $4.90\,\rm m^2$. Assume that the exhaust static pressure and the static pressure acting on the outside of the engine nacelle matches the ambient pressure. The fuel mass flow can be neglected.
  
  
  a) What is the mass flow rate through the engine? [20%]
b) What are the absolute and relative velocities at inlet and outlet? [32%]
c) Calculate the (gauge) static pressure at the engine intake. (Hint: Bernoulli’s equation is for steady state problems - does this imply using absolute or relative velocities in this case?) [24%]
d) How much thrust is produced by the engine? [24%]

 
I have come up with the answers to parts a and b as shown in the image above. However, in part c I can't seem to find the answer, even though it only involves fitting numbers into Bernoulli's equation
$$P + \frac{\rho u^2}{2} + \rho gz = \mathrm{const}$$
Is this because I am using the wrong relative velocity?
 A: First of all you should not be using the classic (incompressible) Bernoulli Equation for this kind of problem as you are clearly in the compressible flow regime. You should be using the isentropic flow relations because they are more physically accurate ($\gamma=1.4$):
$\frac{P_T}{P}=[1+\frac{\gamma -1}{2}M^2]^\frac{\gamma}{\gamma-1}$;   $\frac{T_T}{T}=1+\frac{\gamma -1}{2}M^2$; $\frac{\rho_T}{\rho}=[1+\frac{\gamma -1}{2}M^2]^\frac{1}{\gamma-1}$
plus we will employ the ever-useful adiabatic relation (which is true even for non-isentropic flows):
$T_T=T+\frac{V^2}{2C_p} \Rightarrow \boxed{T=T_T-\frac{V^2}{2C_p}}$
Since you already know the engine massflow rate, the rest of the problem is straightforward. Applying Ideal Gas Law and the definition of Mach number to the basic massflow equation, we have...
$\dot{m}_1=\rho_1 V_1 A_1=(\frac{P_1}{RT_1})(M_1\sqrt{\gamma RT_1})A_1=P_1A_1M_1\sqrt{\frac{\gamma}{RT_1}}$
This last bit is the key, since if you know $\dot{m}_1$, $A_1$, $M_1$, and $T_1$ you will easily be able to find $P_1$. And in fact, this really just boils down to finding $T_1$ because you already know the inlet velocity and $M_1=\frac{V_1}{\sqrt{\gamma RT_1}}$. The key insight here is that the stagnation properties (aircraft reference frame) do not change from the freestream ($\infty$) to the inlet lip ($1$). Thus,
$T_T=T_\infty+\frac{V_\infty^2}{2C_p}=T_1+\frac{V_1^2}{2C_p}$,
which implies that
$\boxed{T_1=T_\infty+\frac{V_\infty^2-V_1^2}{2C_p}}$, with $C_p=\frac{\gamma R}{\gamma-1}$
Since we know the ambient static temperature, aircraft flight speed (245m/s) and the inlet flow velocity (215m/s) we can solve for the inlet static temperature. Everything else follows from this information and you should be able to find everything you need for the rest of the problem. Because the flow is decelerating into the engine during cruise, we would expect the inlet static temperature to be slightly higher than the freestream (and it is).
Another way to compute the answer is to again leverage the fact that the stagnation properties do not change as the flow isentropically decelerates into the engine, except that now we will equate the stagnation pressures instead of temperatures:
$P_T=P_\infty[1+\frac{\gamma -1}{2}M_\infty^2]^\frac{\gamma}{\gamma-1}=P_1[1+\frac{\gamma -1}{2}M_1^2]^\frac{\gamma}{\gamma-1}$
or
$\boxed{P_1=P_\infty\left[\frac{2+(\gamma -1)M_\infty^2}{2+(\gamma -1)M_1^2}\right]^\frac{\gamma}{\gamma-1}}$
where 
$M_\infty=\frac{V_\infty}{\sqrt{\gamma RT_\infty}}$ and $M_1=\frac{V_1}{\sqrt{\gamma RT_1}}$
Good luck!!!
P.S. Your diagram is wrong. The two velocities should be pointing in the same direction. The freestream is approaching the aircraft at 245m/s, and the air actually enters the engine at 215m/s (it slows down). Both velocities are relative to the aircraft reference frame and pointed in the same direction. Also, your engine massflow is wrong because it assumes $\rho_1=\rho_\infty$, which is incorrect. You need to use the isentropic relations here as well in a way analogous to the method outlined above.
