Final answer:
To model Farmer Hoglund's corn yield per acre as a function of amount of fertilizer used, the yield function is Y(N) = 30 + N - N²/400, which includes the initial yield without fertilizer and integrates the diminishing marginal product of fertilizer.
Step-by-step explanation:
To answer the student's question, we first need to develop a yield function for Farmer Hoglund's corn production based on the amount of fertilizer he uses. According to the given information, the marginal product of fertilizer is given by the function 1 - (N/200), where N is the number of pounds of fertilizer applied per acre.
Since the farmer gets 30 bushels of corn per acre without any fertilizer, this initial yield will be our starting point. For each additional pound of fertilizer, the increase in yield will be the marginal product of the fertilizer. Hence, the total yield of corn per acre, Y(N), as a function of the amount of fertilizer used can be represented as:
Y(N) = 30 + ∑ (1 - N/200) from N = 0 to N
Here, ∑ represents the sum of all the marginal products from the first pound of fertilizer to the Nth pound. This simplification can be integrated to create a continuous function representing the yield based on the amount of fertilizer:
Y(N) = 30 + N - N²/400
This integrates the marginal product and takes into account the diminishing returns as more fertilizer is used after a certain point, as indicated by the negative quadratic term.