​The total cost​ (in dollars) of manufacturing x auto body frames is ​C(x)=40,000+300x.  (B) Find the marginal average cost at a production level of 200 units

The total cost function given is:

\[
C(x) = 40,000 + 300x
\]

The **average cost** function, $ A(x) $, is found by dividing the total cost by the number of units $ x $:

\[
A(x) = \frac{C(x)}{x} = \frac{40,000 + 300x}{x}
\]

To find the **marginal average cost** at a production level of 200 units, we first differentiate the average cost function with respect to $ x $ to get the marginal average cost function:

\[
A'(x) = \frac{d}{dx} \left( \frac{40,000 + 300x}{x} \right)
\]

We can use the quotient rule to differentiate this function. The quotient rule states that for $ \frac{f(x)}{g(x)} $, the derivative is:

\[
\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}
\]

Here, $ f(x) = 40,000 + 300x $ and $ g(x) = x $. Let's compute the derivative.The marginal average cost function is:

\[
A'(x) = \frac{300}{x} - \frac{300x + 40,000}{x^2}
\]

At a production level of 200 units, the marginal average cost is:

\[
A'(200) = -1 \, \text{dollar per unit}
\]

This means that at 200 units, the average cost decreases by $1 for each additional unit produced.

Do you have any questions or would you like more details?

Here are 5 related questions to explore further:
1. What is the significance of marginal cost in production decisions?
2. How does the total cost change when production increases from 200 to 201 units?
3. What would the average cost per unit be at a production level of 200 units?
4. How would the marginal average cost change at higher production levels?
5. What is the break-even point if the selling price per unit is known?

**Tip:** The marginal average cost helps in optimizing production efficiency by showing how the average cost changes as you increase output.

The total cost​ (in dollars) of manufacturing x auto body frames is ​C(x)=40,000+300x. (B) Find the marginal average cost at a production level of 200 units.

Explore how to find the marginal average cost at a production level of 200 units using the cost function C(x) = 40,000 + 300x. This detailed solution involves using the quotient rule to differentiate the average cost function and is suitable for students in Grades 11-12 studying calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation