The problem involves a production function:

$f(x, y) = 5 \cdot x^{0.5} \cdot y^{0.5}$ where $x \geq 0$ and $y \geq 0$, and a budget constraint. Specifically, the cost of one unit of input $x$ is 3 monetary units, and the cost of one unit of input $y$ is 4 monetary units. The total budget available for both inputs is 500 monetary units.

The goal is to:

1. Write the budget constraint based on the costs and available budget.
2. Optimize the production function subject to the budget constraint to determine the optimal quantities of $x$ and $y$.

Step 1: Formulate the Budget Constraint

Let $x$ be the number of units of input $x$ and $y$ be the number of units of input $y$. The cost of each input is given:

• $x$ costs 3 units per quantity.
• $y$ costs 4 units per quantity.

The total expenditure on these inputs should equal the budget constraint of 500 monetary units:

$3x + 4y = 500$

Step 2: Solve for the Optimal Production Quantities

To maximize production given the budget constraint, we can use the method of Lagrange multipliers. The objective is to maximize the production function:

$L(x, y, \lambda) = 5x^{0.5}y^{0.5} + \lambda (500 - 3x - 4y)$

We will find the first-order conditions by taking partial derivatives with respect to $x$, $y$, and $\lambda$, and setting them to zero.

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