Finding the Exact Cost of Producing the 21st Machine

Oct 13, 2025 by windailys.com 106 views

The Total Cost of Producing Coffee Machines: A Mathematical Analysis

In this article, we will delve into the world of mathematical economics and explore the concept of total cost in the production of coffee machines. The total cost function, denoted by C(x), represents the exact cost of producing x coffee machines. In this case, the total cost function is given by the quadratic equation: C(x) = 1600 + 60x - 0.4x^2. Our objective is to find the exact cost of producing the 21st machine and use marginal cost to determine the rate at which the cost changes.

The total cost function, C(x), is a quadratic function that represents the total cost of producing x coffee machines. The function is given by:

C(x) = 1600 + 60x - 0.4x^2

where x is the number of coffee machines produced.

Finding the Exact Cost of Producing the 21st Machine

To find the exact cost of producing the 21st machine, we need to substitute x = 21 into the total cost function.

C(21) = 1600 + 60(21) - 0.4(21)^2

First, let's calculate the value of 21^2.

(21)^2 = 441

Now, substitute this value back into the equation.

C(21) = 1600 + 60(21) - 0.4(441)

Next, calculate the value of 60(21).

60(21) = 1260

Now, substitute this value back into the equation.

C(21) = 1600 + 1260 - 0.4(441)

Now, calculate the value of 0.4(441).

0.4(441) = 176.4

Now, substitute this value back into the equation.

C(21) = 1600 + 1260 - 176.4

Finally, calculate the value of the expression.

C(21) = 2563.6

Therefore, the exact cost of producing the 21st machine is $2563.60.

Using Marginal Cost to Determine the Rate of Change

Marginal cost is the rate at which the cost changes when the number of units produced changes by one unit. In other words, it is the derivative of the total cost function with respect to the number of units produced.

To find the marginal cost, we need to find the derivative of the total cost function.

C'(x) = d/dx (1600 + 60x - 0.4x^2)

Using the power rule of differentiation, we get:

C'(x) = 60 - 0.8x

Now, we can use the marginal cost to determine the rate at which the cost changes when the number of units produced changes by one unit.

Marginal Cost at x = 21

To find the marginal cost at x = 21, we need to substitute x = 21 into the marginal cost function.

C'(21) = 60 - 0.8(21)

First, calculate the value of 0.8(21).

0.8(21) = 16.8

Now, substitute this value back into the equation.

C'(21) = 60 - 16.8

Finally, calculate the value of the expression.

C'(21) = 43.2

Therefore, the marginal cost at x = 21 is $43.20.

In this article, we have explored the concept of total cost in the production of coffee machines. We have found the exact cost of producing the 21st machine and used marginal cost to determine the rate at which the cost changes. The marginal cost at x = 21 is $43.20, which means that the cost of producing the 21st machine is $43.20 more than the cost of producing the 20th machine.

[1] Total Cost Function. Retrieved from https://en.wikipedia.org/wiki/Total_cost_function
[2] Marginal Cost. Retrieved from https://en.wikipedia.org/wiki/Marginal_cost

What is the total cost function, and how is it used in economics?
How is marginal cost used to determine the rate of change of the total cost function?
What is the marginal cost at x = 21, and what does it mean in the context of the problem?
The Total Cost of Producing Coffee Machines: A Q&A Article

In our previous article, we explored the concept of total cost in the production of coffee machines. We found the exact cost of producing the 21st machine and used marginal cost to determine the rate at which the cost changes. In this article, we will answer some frequently asked questions related to the total cost function and marginal cost.

Q: What is the total cost function, and how is it used in economics?

A: The total cost function, denoted by C(x), represents the exact cost of producing x coffee machines. It is a quadratic function that takes into account the fixed costs, variable costs, and economies of scale. In economics, the total cost function is used to determine the optimal level of production, pricing, and investment decisions.

Q: How is marginal cost used to determine the rate of change of the total cost function?

A: Marginal cost is the rate at which the cost changes when the number of units produced changes by one unit. It is the derivative of the total cost function with respect to the number of units produced. By using marginal cost, we can determine the rate at which the cost changes and make informed decisions about production and pricing.

Q: What is the marginal cost at x = 21, and what does it mean in the context of the problem?

A: The marginal cost at x = 21 is $43.20, which means that the cost of producing the 21st machine is $43.20 more than the cost of producing the 20th machine. This information can be used to determine the optimal level of production and pricing decisions.

Q: How can the total cost function be used to determine the optimal level of production?

A: The total cost function can be used to determine the optimal level of production by finding the point at which the marginal cost equals the marginal revenue. This point represents the optimal level of production, as it maximizes profits and minimizes costs.

Q: What are some common applications of the total cost function in economics?

A: The total cost function has many applications in economics, including:

Determining the optimal level of production and pricing decisions
Evaluating the impact of changes in production costs on profits and revenue
Analyzing the effects of economies of scale on production costs
Developing pricing strategies and investment decisions

Q: How can the marginal cost function be used to determine the rate of change of the total cost function?

A: The marginal cost function can be used to determine the rate of change of the total cost function by finding the derivative of the total cost function with respect to the number of units produced. This derivative represents the rate at which the cost changes and can be used to make informed decisions about production and pricing.

In this article, we have answered some frequently asked questions related to the total cost function and marginal cost. We have discussed the importance of the total cost function in economics, the use of marginal cost to determine the rate of change of the total cost function, and some common applications of the total cost function in economics.

[1] Total Cost Function. Retrieved from https://en.wikipedia.org/wiki/Total_cost_function
[2] Marginal Cost. Retrieved from https://en.wikipedia.org/wiki/Marginal_cost

What are some other applications of the total cost function in economics?
How can the marginal cost function be used to determine the optimal level of production?
What are some common challenges in using the total cost function in economics?