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Let's solve the problem step by step.

### Given:
The total cost function is:
\[
C(x) = 1600 + 70x - 0.4x^2
\]
Where \(x\) represents the number of food processors produced.

### Part (A) - Exact cost of producing the 31st food processor:
To find the exact cost of producing the 31st food processor, we calculate:
\[
C(31) - C(30)
\]

First, calculate \(C(31)\) and \(C(30)\):
\[
C(31) = 1600 + 70(31) - 0.4(31)^2
\]
\[
C(30) = 1600 + 70(30) - 0.4(30)^2
\]

### 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 \(C(x)\). Let’s find \(C'(x)\):
\[
C'(x) = 70 - 0.8x
\]

We approximate the cost of producing the 31st food processor by evaluating the marginal cost at \(x = 30\):
\[
C'(30) = 70 - 0.8(30)
\]

Let me calculate these values.### Part (A) - Exact cost of producing the 31st food processor:
\[
C(31) = 3385.6 \quad \text{and} \quad C(30) = 3340.0
\]
Thus, the exact cost of producing the 31st food processor is:
\[
C(31) - C(30) = 3385.6 - 3340.0 = 45.6 \, \text{dollars}
\]

### Part (B) - Approximation using the marginal cost:
The marginal cost at \(x = 30\) is:
\[
C'(30) = 46 \, \text{dollars}
\]
Therefore, the marginal cost approximation for producing the 31st food processor is 46 dollars.

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Would you like more details on any part of the solution? Here are some related questions:

1. What is the meaning of marginal cost in this context?
2. Why does the exact cost differ from the marginal cost approximation?
3. How does the shape of the cost function affect the marginal cost?
4. Can we find the marginal cost for any other number of units?
5. How would the production cost change if the quadratic term in the cost function was larger?

**Tip:** The marginal cost provides a good approximation for small changes in production, but the exact cost may vary due to nonlinear effects in the cost function.

The total cost (in dollars) of producing x food processors is C(x) = 1600 + 70x - 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 calculate the exact cost and use the marginal cost to approximate the cost of producing the 31st food processor, using the function C(x) = 1600 + 70x - 0.4x^2. Suitable for Grades 10-12 students studying calculus.

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