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:
[tex]C(x, y) = 99x + 84y[/tex]
[tex]P(x, y) = 100x^{0.25}y^{0.75}[/tex]
For the minimum cost:
[tex]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[/tex]
Using the langrage multiplier,
[tex]99=0.25(100)x^{-0.75}y^{0.75}\lambda...(1\\84=0.25(100)x^{0.25}y^{-0.25}\lambda...(2[/tex]
Dividing both equation 1 and 2
[tex]y=\frac{33}{28}x[/tex]
Substituting [tex]y=\frac{33}{28}x[/tex] in [tex]100x^{0.25}y^{0.75}=60000[/tex], we get:
[tex]x_{min}=530.44[/tex] labor units
Substituting [tex]x_{min}=530.44[/tex] labor units in [tex]y=\frac{33}{28}x[/tex], we get
[tex]y_{min}=\frac{33}{28}*530.44=625.16[/tex] labor unit
[tex]C(x, y) = 99x + 84y=99(530.44)+84(635.16) = $105867\\[/tex]
The minimum cost of producing 60000 units of a product is $105867
