Suppose the process of producing lightweight parkas by​ Polly's Parkas is described by the function
q=20K0.8(L−40)0.2
where q is the number of parkas​ produced, K the number of computerized​ stitching-machine hours, and L the number of​ person-hours of labor. In addition to capital and​ labor,
​$40
worth of raw materials is used in the production of each parka.
Part 2
Note that
∂q∂K=20(0.8)K−0.20(L−40)0.2
and
∂q∂L=20K0.8(0.2)(L−40)−0.80.
Part 3
By minimizing cost subject to the production​ function, derive the​ cost-minimizing demands for K and L as a function of output​ (q), wage rates​ (w), and rental rates on machines​ (r).