Questions: Custom Office makes a line of executive desks. It is estimated that the total cost for making x units of their Senior Executive model is represented by the following function, where C(x) is measured in dollars/year. C(x)=104 x+80000 (a) Find the average cost function C̄. C̄(x)= (b) Find the marginal average cost function C̄ ' C̄'(x)= (c) What happens to C̄(x) when x is very large? lim x → ∞ C̄(x)= Interpret your results. This value is what the production level approaches if the average cost per unit is very high. This value is what the average cost per unit approaches if the production level is very high. This value is what the average cost per unit approaches if the production level is very low. This value is what the production level approaches if the average cost per unit is very low.

Custom Office makes a line of executive desks. It is estimated that the total cost for making x units of their Senior Executive model is represented by the following function, where C(x) is measured in dollars/year.

C(x)=104 x+80000

(a) Find the average cost function C̄.

C̄(x)=

(b) Find the marginal average cost function C̄ '

C̄'(x)=

(c) What happens to C̄(x) when x is very large?

lim x → ∞ C̄(x)=

Interpret your results. This value is what the production level approaches if the average cost per unit is very high. This value is what the average cost per unit approaches if the production level is very high. This value is what the average cost per unit approaches if the production level is very low. This value is what the production level approaches if the average cost per unit is very low.

Transcript text: Custom Office makes a line of executive desks. It is estimated that the total cost for making $x$ units of their Senior Executive model is represented by the following function, where $C(x)$ is measured in dollars/year. \[ C(x)=104 x+80000 \] (a) Find the average cost function $\bar{C}$. \[ \bar{C}(x)= \] $\square$ (b) Find the marginal average cost function $\bar{C}$ ' \[ \bar{C}^{\prime}(x)=\square \] (c) What happens to $\bar{C}(x)$ when $x$ is very large? \[ \lim _{x \rightarrow \infty} \bar{C}(x)=\square \] $\square$ Interpret your results. This value is what the production level approaches if the average cost per unit is very high. This value is what the average cost per unit approaches if the production level is very high. This value is what the average cost per unit approaches if the production level is very low. This value is what the production level approaches if the average cost per unit is very low.

Solution

Solution Steps

To solve the given problem, we need to follow these steps:

(a) The average cost function $\bar{C}(x)$ is found by dividing the total cost function $C(x)$ by the number of units $x$.

(b) The marginal average cost function $\bar{C}'(x)$ is the derivative of the average cost function $\bar{C}(x)$ with respect to $x$.

(c) To determine what happens to $\bar{C}(x)$ when $x$ is very large, we need to find the limit of $\bar{C}(x)$ as $x$ approaches infinity.

Step 1: Find the Average Cost Function

The average cost function $\bar{C}(x)$ is found by dividing the total cost function $C(x)$ by the number of units $x$: \[ \bar{C}(x) = \frac{C(x)}{x} = \frac{104x + 80000}{x} = 104 + \frac{80000}{x} \]

Step 2: Find the Marginal Average Cost Function

The marginal average cost function $\bar{C}'(x)$ is the derivative of the average cost function $\bar{C}(x)$ with respect to $x$: \[ \bar{C}'(x) = \frac{d}{dx} \left( 104 + \frac{80000}{x} \right) = -\frac{80000}{x^2} \]

Step 3: Determine the Limit of $\bar{C}(x)$ as $x$ Approaches Infinity

To determine what happens to $\bar{C}(x)$ when $x$ is very large, we find the limit of $\bar{C}(x)$ as $x$ approaches infinity: \[ \lim_{x \to \infty} \bar{C}(x) = \lim_{x \to \infty} \left( 104 + \frac{80000}{x} \right) = 104 \]

Final Answer

\[ \boxed{104} \]

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