Suppose that the total cost function, in dollars, for the production of \( x \) units of a product is given by the equation shown below. . \[ C(x)=2,890+75 x+0.1 x^{2} \] Find the equation of the average cost function, \( \bar{C}(x) \). \( \bar{C}(x)= \) (a) Find the instantaneous rate of change of average cost with respect to the number of units produced, at any level of production. (b) Find the level of production at which this rate of change equals zero. \( x=\square \) (c) At the value found in part (b), which of the following is true? The average cost function is greater than the total cost function. O The tangent line to the total cost function is a horizontal line. O The instantaneous rate of change of total cost is zero. The tangent line to the average cost function is a horizontal line.

To find the average cost function, we divide the total cost function by the number of units produced. The average cost function is given by: \[ \bar{C}(x) = \frac{C(x)}{x} = \frac{2,890 + 75x + 0.1x^2}{x} = \frac{2,890}{x} + 75 + 0.1x \] To solve part (a), we need to find the instantaneous rate of change of average cost with respect to \( x \). This involves differentiating \( \bar{C}(x) \): \[ \bar{C}'(x) = -\frac{2,890}{x^2} + 0.1 \] For part (b), we set the derivative equal to zero to find the level of production where the rate of change equals zero: \[ -\frac{2,890}{x^2} + 0.1 = 0 \] \[ 0.1 = \frac{2,890}{x^2} \] \[ x^2 = \frac{2,890}{0.1} = 28,900 \] \[ x = 170 \] Now, regarding part (c), at the value found in part (b), the correct statement is: "The tangent line to the average cost function is a horizontal line." This is because when the derivative is zero, it indicates a point of local extremum, typically leading to a horizontal tangent line. Producing goods has a rich history where efficiency methods evolved over centuries. Take the Industrial Revolution, for example, which marked a shift towards factory systems and mass production, significantly influencing cost calculations and the principles of economic production! In the modern economy, understanding average cost functions isn't just an academic exercise—it's crucial for businesses. Companies use average cost analysis to make pricing decisions, minimize costs, and maximize profits, allowing them to remain competitive in a rapidly changing market!