Final answer:
To determine how fast the daily operating budget is changing, we use implicit differentiation on the Cobb-Douglas production function at a constant production level. The rate of change of the budget with respect to time (dy/dt) is found using the chain rule. Calculations would involve solving for y with respect to x and P, and the specifics of an automobile assembly plant.
Step-by-step explanation:
The student is managing an automobile assembly plant with a Cobb-Douglas production function P=30x⁰.²y⁰.¸, where P is the number of automobiles produced per year, x is the number of employees, and y is the daily operating budget in thousands. We are told that the production level is maintained at 800 automobiles per year and that the number of employees is currently 100 and increasing at a rate of 20 per year. The student wants to know how fast the daily operating budget is changing.
To find the rate at which the daily operating budget changes, we must use implicit differentiation with respect to time on the given production function. Since the production level P is constant at 800, its derivative with respect to time (dP/dt) is zero. Therefore, by taking the derivative with respect to time of both sides of the equation and using the chain rule for derivatives, we will solve for dy/dt, which is the rate at which the daily operating budget changes.
The exact calculations are more advanced and would typically involve solving for y in terms of x and P and then differentiating implicitly to find dy/dt. Since the production function is given for an automobile assembly plant, specific values would need to be used to complete the calculation, which are not provided directly in this question. This illustrates how production functions can vary across different products and industries and highlights the complexity of managing production in relation to labor and budget constraints.