Answer
The plane's airspeed is $272~m/s$
Work Step by Step
The pressure difference between the moving air and the stationary air is $\frac{1}{2}\rho_a~v^2$
This pressure difference is equal to the gauge pressure in the manometer, which is $\rho_m~g~h$
We can equate the expressions for the pressure difference to find the airspeed $v$:
$\frac{1}{2}\rho_a~v^2 = \rho_m~g~h$
$v^2 = \frac{2~\rho_m~g~h}{\rho_a}$
$v = \sqrt{\frac{2~\rho_m~g~h}{\rho_a}}$
$v = \sqrt{\frac{(2)~(13600~kg/m^3)~(9.80~m/s^2)(0.25~m)}{0.90~kg/m^3}}$
$v = 272~m/s$
The plane's airspeed is $272~m/s$.
