Question

3: suppose the cost in dollars of producing x units is given by the function c(x) = 2x² + 6x + 50. find the average cost function and the average marginal cost function. then interpret what each mean.

Explanation:

Step-by-Step Format:

Step1: Calculate the average cost function

The average cost function $\overline{C}(x)$ is given by $\overline{C}(x)=\frac{C(x)}{x}$.
Given $C(x) = 0.2x^{2}+6x + 50$, then $\overline{C}(x)=\frac{0.2x^{2}+6x + 50}{x}=0.2x+6+\frac{50}{x}$.

Step2: Calculate the average - marginal cost function

The average - marginal cost function is the derivative of the average cost function.
Using the power rule: if $y = ax^{n}$, then $y^\prime=anx^{n - 1}$, and for $y=\frac{k}{x}=kx^{-1}$, $y^\prime=-kx^{-2}$.
For $\overline{C}(x)=0.2x+6 + 50x^{-1}$, $\overline{C}^\prime(x)=0.2-50x^{-2}=0.2-\frac{50}{x^{2}}$.

Step3: Interpret the average cost function

The average cost function $\overline{C}(x)=0.2x + 6+\frac{50}{x}$ represents the cost per unit of producing $x$ units. The term $0.2x$ is a variable cost component that increases with the number of units produced, the constant term $6$ is a fixed - cost per unit (in a sense, part of the non - linear breakdown of fixed and variable costs in the average sense), and $\frac{50}{x}$ is the fixed - cost component per unit (since the total fixed cost is $50$ when $x = 1$, and as $x$ increases, the fixed cost per unit decreases).

Step4: Interpret the average - marginal cost function

The average - marginal cost function $\overline{C}^\prime(x)=0.2-\frac{50}{x^{2}}$ represents the rate of change of the average cost with respect to the number of units produced. When $\overline{C}^\prime(x)>0$ (i.e., $0.2-\frac{50}{x^{2}}>0$ or $x>\sqrt{\frac{50}{0.2}}=\sqrt{250}\approx15.81$), the average cost is increasing. When $\overline{C}^\prime(x)<0$ (i.e., $x <\sqrt{250}\approx15.81$), the average cost is decreasing.

Answer:

The average cost function is $\overline{C}(x)=0.2x + 6+\frac{50}{x}$, and the average - marginal cost function is $\overline{C}^\prime(x)=0.2-\frac{50}{x^{2}}$. The average cost function $\overline{C}(x)$ gives the cost per unit for producing $x$ units, with components representing variable and (average) fixed costs. The average - marginal cost function $\overline{C}^\prime(x)$ gives the rate of change of the average cost; when positive, the average cost is increasing, and when negative, the average cost is decreasing.