The Total Cost (in Dollars) Of Producing $x$ Food Processors Is Given By $C(x) = 2000 + 80x - 0.2x^2$.(A) Find The Exact Cost Of Producing The 61st Food Processor.(B) Use The Marginal Cost To Approximate The Cost Of Producing The

Introduction

In this article, we will explore the total cost of producing food processors using a given quadratic function. We will first find the exact cost of producing the 61st food processor and then use the marginal cost to approximate the cost of producing the 61st food processor.

The Total Cost Function

The total cost of producing $x$ food processors is given by the quadratic function:

$$C(x) = 2000 + 80x - 0.2x^2$$

This function represents the total cost of producing $x$ food processors, where $x$ is the number of food processors produced.

Finding the Exact Cost of Producing the 61st Food Processor

To find the exact cost of producing the 61st food processor, we need to substitute $x = 61$ into the total cost function:

$$C(61) = 2000 + 80(61) - 0.2(61)^2$$

Expanding and simplifying the expression, we get:

$$C(61) = 2000 + 4880 - 0.2(3721)$$

$$C(61) = 6880 - 744.2$$

$$C(61) = 6135.8$$

Therefore, the exact cost of producing the 61st food processor is $6135.80.

Using the Marginal Cost to Approximate the Cost of Producing the 61st Food Processor

The marginal cost is the rate of change of the total cost function with respect to the number of food processors produced. It represents the additional cost of producing one more food processor.

To find the marginal cost, we need to take the derivative of the total cost function with respect to $x$:

$$C'(x) = 80 - 0.4x$$

The marginal cost is given by the derivative of the total cost function:

$$MC(x) = C'(x) = 80 - 0.4x$$

To approximate the cost of producing the 61st food processor, we can use the marginal cost at $x = 60$:

$$MC(60) = 80 - 0.4(60)$$

$$MC(60) = 80 - 24$$

$$MC(60) = 56$$

The marginal cost at $x = 60$ is $56. This represents the additional cost of producing one more food processor at $x = 60$.

To approximate the cost of producing the 61st food processor, we can multiply the marginal cost at $x = 60$ by the number of food processors produced:

$$\text{Approximate Cost} = MC(60) \times 1$$

$$\text{Approximate Cost} = 56 \times 1$$

$$\text{Approximate Cost} = 56$$

Therefore, the approximate cost of producing the 61st food processor is $56.

Comparison of Exact and Approximate Costs

The exact cost of producing the 61st food processor is $6135.80, while the approximate cost is $56. The approximate cost is significantly lower than the exact cost.

Conclusion

In this article, we found the exact cost of producing the 61st food processor using the total cost function and approximated the cost using the marginal cost. The exact cost is $6135.80, while the approximate cost is $56. The marginal cost can be used to approximate the cost of producing additional food processors, but it may not be accurate for large values of $x$.

References

• [1] Calculus: Early Transcendentals, James Stewart, 8th edition.
• [2] Principles of Microeconomics, Gregory Mankiw, 7th edition.

Glossary

• Marginal Cost: The rate of change of the total cost function with respect to the number of food processors produced.
• Total Cost Function: A quadratic function that represents the total cost of producing $x$ food processors.
• Derivative: A measure of the rate of change of a function with respect to its input variable.
  The Total Cost of Producing Food Processors: Q&A =====================================================

Introduction

In our previous article, we explored the total cost of producing food processors using a given quadratic function. We found the exact cost of producing the 61st food processor and approximated the cost using the marginal cost. In this article, we will answer some frequently asked questions related to the total cost of producing food processors.

Q: What is the total cost function?

A: The total cost function is a quadratic function that represents the total cost of producing $x$ food processors. It is given by the equation:

$$C(x) = 2000 + 80x - 0.2x^2$$

Q: How do I find the exact cost of producing a certain number of food processors?

A: To find the exact cost of producing a certain number of food processors, you need to substitute the number of food processors into the total cost function. For example, to find the exact cost of producing 61 food processors, you would substitute $x = 61$ into the total cost function:

$$C(61) = 2000 + 80(61) - 0.2(61)^2$$

Q: What is the marginal cost?

A: The marginal cost is the rate of change of the total cost function with respect to the number of food processors produced. It represents the additional cost of producing one more food processor. The marginal cost is given by the derivative of the total cost function:

$$MC(x) = C'(x) = 80 - 0.4x$$

Q: How do I use the marginal cost to approximate the cost of producing a certain number of food processors?

A: To use the marginal cost to approximate the cost of producing a certain number of food processors, you need to find the marginal cost at the desired number of food processors and multiply it by the number of food processors produced. For example, to approximate the cost of producing 61 food processors, you would find the marginal cost at $x = 60$ and multiply it by 1:

$$\text{Approximate Cost} = MC(60) \times 1$$

$$\text{Approximate Cost} = 56 \times 1$$

$$\text{Approximate Cost} = 56$$

Q: Why is the marginal cost not always accurate?

A: The marginal cost is not always accurate because it is an approximation of the total cost function. As the number of food processors produced increases, the marginal cost may not accurately reflect the actual cost of producing additional food processors.

Q: Can I use the marginal cost to find the total cost of producing a certain number of food processors?

A: No, you cannot use the marginal cost to find the total cost of producing a certain number of food processors. The marginal cost is a measure of the rate of change of the total cost function, not the total cost itself.

Q: What is the relationship between the total cost function and the marginal cost?

A: The total cost function and the marginal cost are related in that the marginal cost is the derivative of the total cost function. This means that the marginal cost represents the rate of change of the total cost function with respect to the number of food processors produced.

Conclusion

In this article, we answered some frequently asked questions related to the total cost of producing food processors. We hope that this article has provided you with a better understanding of the total cost function and the marginal cost.

References

• [1] Calculus: Early Transcendentals, James Stewart, 8th edition.
• [2] Principles of Microeconomics, Gregory Mankiw, 7th edition.

Glossary

• Marginal Cost: The rate of change of the total cost function with respect to the number of food processors produced.
• Total Cost Function: A quadratic function that represents the total cost of producing $x$ food processors.
• Derivative: A measure of the rate of change of a function with respect to its input variable.