The total cost (in dollars) of producing x food processors is
𝐶(𝑥) = 2000 + 50𝑥 − 0.5𝑥
2
(A) Find the exact cost of producing the 21st food processor.
(B) Use marginal cost to approximate the cost of producing the 21st food processor.
(C) Find the average cost and the marginal average cost of the given C(x). Interpret the average cost
and marginal average cost at 21 food processors.

We are given the total cost function for producing $ x $ food processors:

\[
C(x) = 2000 + 50x - 0.5x^2
\]

Let's go through each part of the problem step-by-step.

### (A) Find the exact cost of producing the 21st food processor.

The exact cost of producing the 21st food processor is the difference between the total cost of producing 21 processors and the total cost of producing 20 processors:

\[
C(21) - C(20)
\]

First, calculate $ C(21) $:

\[
C(21) = 2000 + 50(21) - 0.5(21)^2 = 2000 + 1050 - 220.5 = 2829.5
\]

Now, calculate $ C(20) $:

\[
C(20) = 2000 + 50(20) - 0.5(20)^2 = 2000 + 1000 - 200 = 2800
\]

Thus, the exact cost of producing the 21st food processor is:

\[
C(21) - C(20) = 2829.5 - 2800 = 29.5
\]

So, the exact cost of producing the 21st food processor is **$29.50**.

### (B) Use marginal cost to approximate the cost of producing the 21st food processor.

The marginal cost $ C'(x) $ is the derivative of the total cost function $ C(x) $. We first differentiate $ C(x) $:

\[
C'(x) = \frac{d}{dx} \left( 2000 + 50x - 0.5x^2 \right) = 50 - x
\]

Now, to approximate the cost of producing the 21st food processor, evaluate the marginal cost at $ x = 20 $ (since marginal cost at $ x $ approximates the cost of producing the next unit):

\[
C'(20) = 50 - 20 = 30
\]

Thus, the marginal cost approximation for producing the 21st food processor is **$30**.

### (C) Find the average cost and the marginal average cost of $ C(x) $. Interpret the average cost and marginal average cost at 21 food processors.

1. **Average cost**: The average cost $ \text{AC}(x) $ is the total cost divided by the number of units produced:

\[
\text{AC}(x) = \frac{C(x)}{x} = \frac{2000 + 50x - 0.5x^2}{x}
\]

Simplifying this expression:

\[
\text{AC}(x) = \frac{2000}{x} + 50 - 0.5x
\]

Now, evaluate the average cost at $ x = 21 $:

\[
\text{AC}(21) = \frac{2000}{21} + 50 - 0.5(21) = 95.24 + 50 - 10.5 = 134.74
\]

So, the average cost at 21 food processors is **$134.74**.

2. **Marginal average cost**: The marginal average cost is the derivative of the average cost function:

\[
\text{MAC}(x) = \frac{d}{dx} \left( \frac{C(x)}{x} \right) = -\frac{2000}{x^2} - 0.5
\]

Now, evaluate the marginal average cost at $ x = 21 $:

\[
\text{MAC}(21) = -\frac{2000}{21^2} - 0.5 = -\frac{2000}{441} - 0.5 = -4.54 - 0.5 = -5.04
\]

So, the marginal average cost at 21 food processors is **$-5.04**.

### Interpretation:

- The **average cost** at 21 processors is $134.74, meaning that on average, it costs this amount to produce each of the 21 processors.
- The **marginal average cost** of -5.04 means that, as we increase production to 21 processors, the average cost decreases by approximately $5.04 per unit.

---

Do you want details or have any questions about these steps?

### Related Questions:
1. How does the total cost function affect pricing strategy in economics?
2. What is the difference between marginal cost and average cost in long-term production?
3. How can marginal cost help in making production decisions in manufacturing?
4. What role does the marginal cost play when optimizing production levels?
5. How can average and marginal cost analysis be used to predict profitability?

### Tip:
When working with marginal cost, it is important to remember that it represents the cost of producing *one additional unit*, and it helps businesses decide the optimal level of production to maximize profit.

The total cost (in dollars) of producing x food processors is C(x) = 2000 + 50x − 0.5x^2. (A) Find the exact cost of producing the 21st food processor. (B) Use marginal cost to approximate the cost of producing the 21st food processor. (C) Find the average cost and the marginal average cost of the given C(x). Interpret the average cost and marginal average cost at 21 food processors.

This detailed analysis covers how to find the exact cost of producing the 21st food processor using C(x) = 2000 + 50x - 0.5x^2. Learn how to apply marginal cost approximations, average cost, and marginal average cost analysis for 21 units of production. Suitable for undergraduate students studying calculus or economics.

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