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

Answer:

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

Step-by-step 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:

$100x^{0.25}y^{0.75}=60000\\x^{0.25}(\frac{33x}{28} )^{0.75}=600\\x^{0.25}(\frac{33}{28} )^{0.75}x^{0.75}=600\\x^{0.25+0.75}*1.131=600$

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

$y=\frac{33}{28}x$

Substituting $y=\frac{33}{28}x$ in $100x^{0.25}y^{0.75}=60000$, we get:

$x_{min}=530.44$ labor units

Substituting $x_{min}=530.44$  labor units in $y=\frac{33}{28}x$, we get

$y_{min}=\frac{33}{28}*530.44=625.16$ 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