Maximizing output

[TheNextOne]

A company uses two kinds of raw materials, copper and zinc, to manufacture a product. The number of products that can be made from x kilograms of copper and y kilograms of zinc is
Q = xy

One kilogram of copper costs $20, and one kilogram of zinc costs $10. If the manufacturer has a budget of $10,000, how many kilograms of each raw material should the manufacturer purchase in order to maximize the number of products that can be made?

I used laragnge multipliers for this question.

My equation was xy - z(20x+10y-10,000).
I found the first derivatives of each value and got z = 25, x = 250 and y = 500. Is this correct? Thank you.

 

[galactus]

You can do it with Lagrange or without. BTW, what's the z for?.

Anyway, \(\displaystyle 20x+10y=10000\)

Solve for \(\displaystyle y=1000-2x\)

Sub into \(\displaystyle Q=xy\):

\(\displaystyle Q=x(1000-2x)=1000x-2x^{2}\)

Differentiate:

\(\displaystyle \frac{dQ}{dx}=1000-4x\)

Set to 0 and solve for x:

\(\displaystyle 1000-4x=0\)

Looks like x=250. Therefore, y=500.