This problem is solved using the method of Lagrange multipliers. The Lagrangian is:
Φ = 8K + 2L + λ(K.5L.5 - 144)
Differentiating with respect to K, L and
Φ/∂K = 8 + λ(.5L.5/K.5)
Φ/∂L = 2 + λ(.5K.5/L.5)
= K.5L.5 - 144
Setting these derivatives equal to zero and solving for K, L and
K = 72, L = 288, and TC = 1,152.
If K = 16, then Q = 4L.5.
Thus, for Q = 144, L = 1,296 and TC = 2,720.