Question

{Use of Tech} A different interpretation of marginal cost Suppose a large company makes 25,000 gadgets per year in batches of x items at a time. After analyzing setup costs to produce each batch and taking into account storage costs, planners have determined that the total cost C(x) of producing 25,000 gadgets in batches of x items at a time is given by C(x) = 1,250,000+125,000,000 / x + 1.5x.
a. Determine the marginal cost and average cost functions. Graph and interpret these functions.

Accepted Answer

First, let's understand the concept of marginal cost. Marginal cost is the derivative of the total cost function C(x) with respect to x. It represents the rate of change of the total cost as the number of items per batch changes. To find the marginal cost function, differentiate the given total cost function C(x) = 1,250,000 + 125,000,000 / x + 1.5x with respect to x. Use the power rule and the derivative of 1/x, which is $$-1/x^2. $$Next, let's determine the average cost function. The average cost function is the total cost C(x) divided by the number of items produced, which is 25,000. So, the average cost function A(x) is A(x) = C(x) / 25,000. To graph these functions, plot the marginal cost function and the average cost function on the same set of axes. The x-axis represents the number of items per batch, and the y-axis represents the cost. Analyze how the marginal cost changes as x increases and how the average cost behaves. Interpret the graphs: The marginal cost graph shows how the cost changes with each additional item per batch. If the marginal cost is decreasing, it indicates economies of scale. The average cost graph helps understand the cost per item produced, which is useful for pricing and production decisions.

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Concetti chiave

Marginal Cost

Average Cost

Graphing Functions