You have 2 unknowns so you need to find 2 simultaneous equations and solve them.

There are 2 ways to do this, the long way and the short way.

The long way to do this is to set up a lagrangian. You want to minimise the total cost, subject to the constraint that you have produced 121000 airframes.

Total cost: K + 10L

Constraint: \(\displaystyle 121000 = (\sqrt{L} + \sqrt{K})^{2}\)

Lagrangian

\(\displaystyle Z = K + 10L -\lambda ((\sqrt{L} + \sqrt{K})^{2} -121000)\)

If you solve this in the usual way, you will get 2 simultaneous equations:

\(\displaystyle 121000 = (\sqrt{L} + \sqrt{K})^{2}\)

\(\displaystyle \frac{MPL}{10} = \frac{MPK}{1}\)

Solve those to get your answers of L and K.

** short cut**

You only need the ratio L/K. You can get this from the second simultaneous equation on its own

\(\displaystyle \frac{MPL}{10} = \frac{MPK}{1}\)

\(\displaystyle \frac{MPL}{MPK} = \frac{10}{1}\)

\(\displaystyle \frac{MPL}{MPK} = 10\)

\(\displaystyle \frac{(K^{0.5} + L^{0.5})L^{-0.5}}{(K^{0.5} + L^{0.5})K^{-0.5}} = 10\)

\(\displaystyle \frac{L^{-0.5}}{K^{-0.5}} = 10\)

\(\displaystyle \frac{K^{0.5}}{L^{0.5}} = 10\)

\(\displaystyle \frac{K}{L} = 100\)

Interestingly, for this production function, the ratio of K/L is constant at all levels of output. So you didn't need to know that there were 121000 airframes after all.