Find the minimum cost of producing 20,000 units of a product, where x is the number of units of labor (at$48 per unit) and y is the number of units of capital (at $36 per unit).

Question

P(x,y)=100x^.6y^.4

Answer 1

I do not know if you have done Lagrange multipliers. Here's a non-linear application.

The basic ideas can be found in your textbook, or you can refer to a previous post:
http://www.jiskha.com/display.cgi?id=1291776389

We will apply the method here.

The production function, P(x,y) represents the number of units produced when the quantities of labour (x) and capital (y) are available, where
P(x,y)=100x^0.6y^0.4
This is our constraint equation, since the number of units produced must equal 20000, so
100x^0.6y^0.4-20000=0 ... (1)

For the resources, we would like to minimize the total cost, which can be represented by the cost function,
C(x,y)=48x+36y ....(2)

For C(x,y) to be a minimum, subject to the constraint equation (1), we form a new function,
Λ(x,y)=C(x,y)+λ(P(x,y)-20000) ....(3)

By partial differentiation of (3) with respect to x and y, we obtain equations

∂Λ/∂x
=48 - 60λx^(-0.4)y^(0.4) = 0 ...(4)
∂Λ/∂y
=48 - 40λx^(0.6)y^(-0.6) = 0 ... (5)
and finally the constraint equation (1)
100x^0.6y^0.4-20000=0 ... (6)

Solve algebraic equations (4), (5) and (6) for x,y and λ using trial and error, or other iterative methods, we obtain (this is the most tedious part of the problem):
x=209.648
y=186.354
λ does not enter into our calculations, but it equals 0.8385925

Total cost
=C(209.648,186.354)
=$16771.85

Check:
No. of units produced
=P(209.648,186.354)
=19999.99

Note: I did not make integral numbers of the resources, as it was not indicated in the question that these quantities are integers. In any case, the required quantity could be a multiple of 20000 in which case the fractional values could apply.