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Let's go through the question step by step.

The total cost function is given as:
\[
C(x) = 1900 + 60x - 0.4x^2
\]
where $C(x)$ is the total cost (in dollars) of producing $x$ food processors.

### Part (A): Find the exact cost of producing the 31st food processor.

The exact cost of producing the 31st food processor is found by calculating the difference between the total cost of producing 31 processors and 30 processors:
\[
\text{Exact cost} = C(31) - C(30)
\]

First, let's compute $C(31)$ and $C(30)$:
\[
C(31) = 1900 + 60(31) - 0.4(31)^2
\]
\[
C(30) = 1900 + 60(30) - 0.4(30)^2
\]

### Part (B): Use the marginal cost to approximate the cost of producing the 31st food processor.

The marginal cost is the derivative of the cost function, which approximates the cost of producing one additional unit. We need to calculate the derivative of $C(x)$, which is:
\[
C'(x) = 60 - 0.8x
\]

To approximate the cost of producing the 31st food processor, we evaluate the marginal cost at $x = 30$:
\[
\text{Approximate cost} = C'(30) = 60 - 0.8(30)
\]

Let me calculate these values now.### Part (A): The exact cost of producing the 31st food processor is approximately \$35.60.

### Part (B): Using the marginal cost, the approximate cost of producing the 31st food processor is \$36.00.

Would you like any further details or clarification on these calculations?

Here are 5 related questions to extend your understanding:
1. How would the exact cost change if we were to calculate for the 32nd processor?
2. What happens to the marginal cost as the number of units produced increases?
3. Why is the approximate cost close to the exact cost?
4. How would a higher degree term in the cost function (e.g., $x^3$) affect the calculations?
5. Can we calculate the total cost for producing 50 processors using this method?

**Tip:** The marginal cost helps provide a quick approximation, especially when exact calculations are more complex!

The total cost (in dollars) of producing x food processors is C(x) = 1900 + 60x - 0.4x^2. (A) Find the exact cost of producing the 31st food processor. (B) Use the marginal cost to approximate the cost of producing the 31st food processor.

Learn how to find the exact and marginal cost of producing the 31st food processor using the cost function C(x) = 1900 + 60x - 0.4x^2. Understand how calculus helps in determining the marginal cost approximation and exact cost difference. Suitable for Grades 11-12 and college students.

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