The total cost (in dollars) of producing x food
processors is

C1x2 = 2,000 + 50x - 0.5x2
(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.

Given the total cost function for producing $ x $ food processors:

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

we'll proceed to answer the two parts of this question.

### Part (A): Exact Cost of Producing the 21st Food Processor

The exact cost of producing the $ n $-th food processor is calculated by finding the difference in the total cost function between producing $ n $ and $ n-1 $ units. Thus, to find the exact cost of producing the 21st food processor, we calculate:

\[
\text{Exact Cost of 21st processor} = C(21) - C(20)
\]

1. **Calculate $ C(21) $:**

   \[
   C(21) = 2000 + 50(21) - 0.5(21)^2
   \]
   \[
   = 2000 + 1050 - 0.5 \times 441
   \]
   \[
   = 2000 + 1050 - 220.5
   \]
   \[
   = 2829.5
   \]

2. **Calculate $ C(20) $:**

   \[
   C(20) = 2000 + 50(20) - 0.5(20)^2
   \]
   \[
   = 2000 + 1000 - 0.5 \times 400
   \]
   \[
   = 2000 + 1000 - 200
   \]
   \[
   = 2800
   \]

3. **Exact Cost of the 21st Processor:**

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

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

---

### Part (B): Approximate Cost Using Marginal Cost

The marginal cost, $ C'(x) $, is the derivative of the total cost function $ C(x) $. We can approximate the cost of producing the 21st food processor by evaluating the marginal cost at $ x = 20 $.

1. **Find $ C'(x) $:**

   \[
   C(x) = 2000 + 50x - 0.5x^2
   \]
   Differentiating with respect to $ x $:
   \[
   C'(x) = 50 - x
   \]

2. **Evaluate $ C'(x) $ at $ x = 20 $:**

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

So, the approximate cost of producing the 21st food processor, using the marginal cost, is **$30**.

---

### Summary of Answers:
- **(A) Exact Cost of the 21st Food Processor:** $29.50
- **(B) Approximate Cost Using Marginal Cost:** $30

Would you like more details, or have any questions?

### Further Questions for Practice:
1. How would you calculate the cost of producing the 22nd food processor exactly?
2. What would the marginal cost be if the production increased to 25 processors?
3. How does the marginal cost function indicate the trend in production costs?
4. What is the significance of the negative term $ -0.5x^2 $ in the cost function?
5. How could you use the marginal cost function to estimate costs over a range of units produced?

### Tip:
Marginal cost is most accurate for estimating the cost of producing one additional unit when evaluated close to the desired quantity, but as quantities increase, the estimate can diverge from the exact incremental cost.

The total cost (in dollars) of producing x food processors is C(x) = 2000 + 50x - 0.5x^2. Find the exact cost of producing the 21st food processor and use marginal cost to approximate the cost.

Explore how to calculate the exact cost and approximate cost using marginal cost for producing the 21st food processor based on the cost function C(x) = 2000 + 50x - 0.5x^2. Detailed solution included.

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