Problem 13-3 Let S represent the amount of steel produced (in tons). Steel produ

Question

Problem 13-3 Let S represent the amount of steel produced (in tons). Steel production is related to the amount of labor used (L) and the amount of capital used (C) by the following function S = 15 L0.2 C 0.8 In this formula L represents the units of labor input and C the units of capital input. Each unit of labor costs $50, and each unit of capital costs $120 (a) Formulate an optimization problem that will determine how much labor and capital are needed in order to produce 55,000 tons of steel at minimum cost. Min 50 120 15 L02 c 0.8 150 2 0 (b) Solve the optimization problem you formulated in part a. Hint: When using Excel Solver, start with an initial L>0 andC>0 If required, round your answers to two decimal places 1914.5 3,829 478,625 Cost =

Explanation / Answer

We have to minimize the costs and maximize the prdocution.

S= 15 L0.2 C0.8

We have to minimize

C = 50 L + 120 C

Here,

15 L0.2 C0.8 = 55000

15 L0.2 C0.8 - 55000 = 0

so the optimization equation or may be solving with langrange

F(L,C) = 50L + 120 C - k(15 L0.2 C0.8 - 55000)

so for optimized cost

df/dL = 0 = df/dC

dF/dL = 50 - 3k L-0.8 = 0

50 = 3k L-0.8

dF/dC = 120 - 12kC-0.2 = 0

10 = k C-0.2

by solving this

we get

L = $ 2436.65

C = $ 4061.08

Cost = $ 609161.49

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