In Baggs, Wyoming, cattle can be produced according to the following process: C = G/10 + P/40, where C is the number of cattle, G are bushels of grain, and P are acres of pasture.If grain costs $4 per bushel and pasture costs $4 per acre, how many cattle can Rancher Roy produce with a budget of $4,000?

Question

Quizplus · Accepted Answer

We can use the budget equation to solve for the maximum number of cattle:
$4G + $4P = $4000
G + P = 1000
Substitute the production function into the budget equation:
C F1F1F1S1F1F1F10 F1F1F1S1F1F1F10 G/10 F1F1F1S1F1F1F10 P/40
C (F1F1F1S1F1F1F10 G/10 F1F1F1S1F1F1F10 P/40) = C(0.1G + 0.025P)
We want to maximize C subject to the constraint G + P = 1000 and the budget constraint.
We can use Lagrange multipliers to solve for the maximum:
L(C, G, P, λ) = C(0.1G + 0.025P) - λ($4G + $4P - $4000) - μ(G + P - 1000)
Taking partial derivatives and setting them equal to zero, we get:
0.1C - 4λ - μ = 0
0.025C - 4λ - μ = 0
4G + 4P - 4000 = 0
G + P - 1000 = 0
Solving this system of equations gives us C = 100, G = 600, P = 400.
Therefore, Rancher Roy can produce 100 cattle with a budget of $4,000.

In Baggs, Wyoming, Cattle Can Be Produced According to the Following