Economist Henry Schultz devised the following demand...

Economist Henry Schultz devised the following demand function for corn: $$p=\frac{6,570,000}{q^{1.3}}$$ where $q$ is the number of bushels of corn that could be sold at $p$ dollars per bushel in one year. ${ }^{7}$ Assume that at least 10,000 bushels of corn per year must be sold. a. How much should farmers charge per bushel of corn to maximize annual revenue? b. How much corn can farmers sell per year at that price? c. What will be the farmers' resulting revenue?

Economist Henry Schultz devised the following demand function for corn:
$$p=\frac{6,570,000}{q^{1.3}}$$
where $q$ is the number of bushels of corn that could be sold at $p$ dollars per bushel in one year. ${ }^{7}$ Assume that at least 10,000 bushels of corn per year must be sold.
a. How much should farmers charge per bushel of corn to maximize annual revenue?
b. How much corn can farmers sell per year at that price?
c. What will be the farmers' resulting revenue?

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Key Concepts

Differentiation

Differentiation is the mathematical process used to compute the derivative of a function. It is critical in optimization problems as it helps identify critical points where the function's rate of change is zero, which may correspond to maximum or minimum values.

Optimization

Optimization involves finding the maximum or minimum of a function. In the context of revenue maximization, it requires determining the quantity that maximizes the revenue function, taking into account the relationship described by the demand function.

Constraint Analysis

Constraint analysis addresses any limitations or bounds imposed on an optimization problem. In scenarios where a minimum or maximum value is required (such as a minimum number of units sold), it ensures that the solution is not only optimal but also feasible within the specified conditions.

Demand Function

The demand function represents the relationship between the price at which a product is sold and the quantity demanded. It indicates how price varies with the amount sold and is fundamental in understanding market behavior.

Revenue Function

The revenue function is defined as the product of the price per unit and the quantity sold. It encapsulates the total income from sales and serves as the objective function to be optimized in revenue maximization problems.

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