Question

P(x,y)=2500x

(
3/5)y
(
2/5)

(a) Find the number of units produced with 25 units of labor and 1064 units of capital. p = units (Round to the nearest whole number.)

(b) Find... (the rest of the text is not provided, but it seems like you have more to include).

Answer 1

(a) The production is about 384 units with 25 units of labor and 1064 units of capital. (b) Maximizing the production function
`\(P(x, y) = 2500x \left((3)/(5)\right)^y \left((2)/(5)\right)\)` using 25 units of labor and 1064 units of capital yields an optimal output of 384 units.

Step-by-step explanation:

In this production scenario defined by the function
`\(P(x, y) = 2500x \left((3)/(5)\right)^y \left((2)/(5)\right)\)` , the goal is to determine the optimal output given 25 units of labor and 1064 units of capital. The function incorporates the contributions of both labor and capital with corresponding exponents. Upon substituting the provided values into the equation, specifically
`\(x = 25\)` and
`\(y = 1064\)` , the calculation yields an output of approximately 384 units, representing the maximum production achievable with the given resources.

The exponents
`\((3)/(5)\)` and
`\((2)/(5)\)` in the function highlight the proportional impact of labor and capital on the overall output. The optimization process involves finding the combination of these inputs that maximizes the production function. In this case, employing 25 units of labor and 1064 units of capital results in the peak production level of 384 units.

This analysis emphasizes the significance of balancing labor and capital inputs to achieve the most efficient and productive outcome, as indicated by the provided production function.