Questions: The total cost (in dollars) of manufacturing x auto body frames is C(x)=80,000+300x. (A) Find the average cost per unit if 800 frames are produced. [Hint: Recall that C(x) is the average cost per unit] (B) Find the average value of the cost function over the interval [0,800]. (C) Discuss the difference between parts (A) and (B).

The total cost (in dollars) of manufacturing x auto body frames is C(x)=80,000+300x.
(A) Find the average cost per unit if 800 frames are produced. [Hint: Recall that C(x) is the average cost per unit]
(B) Find the average value of the cost function over the interval [0,800].
(C) Discuss the difference between parts (A) and (B).

Transcript text: The total cost (in dollars) of manufacturing $x$ auto body frames is $C(x)=80,000+300 x$. (A) Find the average cost per unit if 800 frames are produced. [Hint: Recall that $\overline{\mathrm{C}}(\mathrm{x})$ is the average cost per unit] (B) Find the average value of the cost function over the interval $[0,800]$. (C) Discuss the difference between parts (A) and (B).

Solution

Solution Steps

To solve the given problems, we will follow these steps:

(A) To find the average cost per unit when 800 frames are produced, we will use the formula for average cost per unit, which is the total cost divided by the number of units produced.

(B) To find the average value of the cost function over the interval [0, 800], we will calculate the definite integral of the cost function over this interval and then divide by the length of the interval.

(C) We will discuss the conceptual difference between the average cost per unit for a specific number of units and the average value of the cost function over an interval.

Step 1: Average Cost per Unit for 800 Frames

To find the average cost per unit when 800 frames are produced, we use the formula:

\[ \overline{C}(x) = \frac{C(x)}{x} \]

Substituting $C(800) = 80000 + 300 \cdot 800$:

\[ C(800) = 80000 + 240000 = 320000 \]

Thus, the average cost per unit is:

\[ \overline{C}(800) = \frac{320000}{800} = 400 \]

Step 2: Average Value of the Cost Function over [0, 800]

To find the average value of the cost function $C(x)$ over the interval $[0, 800]$, we calculate:

\[ \text{Average Value} = \frac{1}{b-a} \int_a^b C(x) \, dx \]

where $a = 0$ and $b = 800$. The cost function is:

\[ C(x) = 80000 + 300x \]

Calculating the definite integral:

\[ \int_0^{800} (80000 + 300x) \, dx = \left[ 80000x + \frac{300x^2}{2} \right]_0^{800} \]

Evaluating this gives:

\[ = 80000 \cdot 800 + 150 \cdot 800^2 = 64000000 + 96000000 = 160000000 \]

Now, dividing by the length of the interval:

\[ \text{Average Value} = \frac{160000000}{800} = 200000 \]

Step 3: Discussion of Differences

The average cost per unit for 800 frames is $400$, which reflects the cost per unit when producing exactly 800 frames. In contrast, the average value of the cost function over the interval $[0, 800]$ is $200000$, representing the average cost across all production levels from 0 to 800 frames. This indicates that the average cost per unit can vary significantly depending on the production level.