Question

Find the minimum cost of producing 60000 units of a product, where x is the number of units of labor, at $99 per unit, and y is the number of units of capital expended, at $84 per unit. And determine how many units of labor and how many units of capital a company should use. Where the production level is given by P(x, y) = 100x^0.25 y^0.75

Answer 1

The minimum cost of producing 60000 units of a product is $105867

Explanation:

Since x is the number of units of labor, at $99 per unit, and y is the number of units of capital expended, at $84 per unit, The cost C(x,y) is given by:

`C(x, y) = 99x + 84y`

`P(x, y) = 100x^(0.25)y^(0.75)`

For the minimum cost:

`100x^(0.25)y^(0.75)=60000\\x^(0.25)((33x)/(28) )^(0.75)=600\\x^(0.25)((33)/(28) )^(0.75)x^(0.75)=600\\x^(0.25+0.75)*1.131=600`

Using the langrage multiplier,

`99=0.25(100)x^(-0.75)y^(0.75)\lambda...(1\\84=0.25(100)x^(0.25)y^(-0.25)\lambda...(2`

Dividing both equation 1 and 2

`y=(33)/(28)x`

Substituting
`y=(33)/(28)x` in
`100x^(0.25)y^(0.75)=60000`, we get:

`x_(min)=530.44` labor units

Substituting
`x_(min)=530.44` labor units in
`y=(33)/(28)x`, we get

`y_(min)=(33)/(28)*530.44=625.16` labor unit

`C(x, y) = 99x + 84y=99(530.44)+84(635.16) = $105867\\`

The minimum cost of producing 60000 units of a product is $105867