Let's break this problem into parts:

Part (A): Exact Cost of Producing the 71st Food Processor

The total cost function is given by:

$C(x) = 2200 + 20x - 0.1x^2$

To find the exact cost of producing the 71st food processor, we need to find the difference in total costs between producing 71 food processors and 70 food processors:

$\text{Cost of the 71st processor} = C(71) - C(70)$

1. Find $C(71)$:

$C(71) = 2200 + 20(71) - 0.1(71^2)$

2. Find $C(70)$:

$C(70) = 2200 + 20(70) - 0.1(70^2)$

After calculating both, subtract the values to get the exact cost of producing the 71st food processor.

Part (B): Approximate Cost Using the Marginal Cost

The marginal cost is the derivative of the cost function $C(x)$. We can approximate the cost of producing the 71st food processor using the marginal cost at $x = 70$.

First, find the derivative of the cost function $C(x)$:

$C'(x) = \frac{d}{dx} \left( 2200 + 20x - 0.1x^2 \right)$

Then evaluate $C'(70)$, which will give us the marginal cost at the 70th processor. The result will approximate the cost of producing the 71st food processor.

Let me calculate both parts now:

Part (A): Exact Cost

$C(71) = 2200 + 20(71) - 0.1(71^2) = 2200 + 1420 - 0.1(5041) = 2200 + 1420 - 504.1 = 4115.9$

$C(70) = 2200 + 20(70) - 0.1(70^2) = 2200 + 1400 - 0.1(4900) = 2200 + 1400 - 490 = 3110$

$\text{Exact Cost of 71st Processor} = C(71) - C(70) = 4115.9 - 3110 = 1005.9$

So, the exact cost of producing the 71st food processor is $1005.90.

Part (B): Approximate Cost Using the Marginal Cost

Now, calculate the marginal cost function:

$C'(x) = \frac{d}{dx} (2200 + 20x - 0.1x^2) = 20 - 0.2x$

Evaluate at $x = 70$:

$C'(70) = 20 - 0.2(70) = 20 - 14 = 6$

So, the approximate marginal cost of producing the 71st food processor is $6.

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Final Answers:

• (A) Exact cost of producing the 71st food processor: $1005.90
• (B) Approximate cost of producing the 71st food processor using the marginal cost: $6

Would you like a further explanation or more details on any of these calculations? Here are some related questions to help you dive deeper:

1. How do you interpret the marginal cost in real-world terms?
2. What would happen if the cost function had a linear form, without the quadratic term?
3. How could the marginal cost approximation be more accurate for larger values of $x$?
4. What does the second derivative of the cost function tell you about the behavior of the cost?
5. How would you use this information for pricing or cost-control decisions in a business?

Tip: The marginal cost is a good approximation when the change in $x$ is small, as it uses the derivative to estimate the rate of change.