The Karman line is a theoretical construct that separates the Earth's atmosphere from outer space. It is defined as the height at which the lift on an aircraft flying at the speed of a polar satellite $(8 \, km/s)$ is equal to its weight. Taking a fighter aircraft of wing area $30 \, m^2$ and mass $7500 \, kg$,the height of the Karman line above the ground will be in the range of .............. $km$. (Assume the density of air at height $h$ above the ground to be $\rho(h) = 1.2 e^{-h/10} \, kg/m^3$,where $h$ is in $km$,and the lift force to be $\frac{1}{2} \rho v^2 A$,where $v$ is the speed of the aircraft and $A$ is its wing area.)

• A
  $25-50$
• B
  $75-100$
• C
  $125-150$
• D
  $175-200$