Analysis of the Function Equation

Oct 13, 2025 by miodeya.com 58 views

Introduction

In this article, we will explore the mathematical concept of function equations and apply it to a real-world scenario. Marco is building a dog house and wants to know the total cost of the project. We will use function equations to represent the total cost and analyze the relationship between the cost of the dog house kit, the number of paint cans used, and the total cost.

The Problem

Marco pays $\$65$ for a dog house kit and $\$12$ for each can of paint, $p$ , he uses to paint the dog house. We need to find a function equation that represents the total cost to build the dog house.

The Function Equation

Let's denote the total cost as $c(p)$ . The cost of the dog house kit is a fixed cost of $\$65$ , and the cost of each can of paint is $\$12$ . Since Marco uses $p$ cans of paint, the total cost of the paint is $12p$ . Therefore, the total cost is the sum of the fixed cost and the cost of the paint:

c(p) = 65 + 12p

This is a linear function equation, where the total cost is a linear function of the number of paint cans used.

Analysis of the Function Equation

Let's analyze the function equation $c(p) = 65 + 12p$ .

Fixed Cost: The fixed cost is $\$65$ , which is the cost of the dog house kit. This cost is independent of the number of paint cans used.
Variable Cost: The variable cost is $\$12p$ , which is the cost of each can of paint. This cost varies linearly with the number of paint cans used.
Total Cost: The total cost is the sum of the fixed cost and the variable cost: $c(p) = 65 + 12p$ .

Graph of the Function Equation

The graph of the function equation $c(p) = 65 + 12p$ is a straight line with a positive slope. The y-intercept is 65, which represents the fixed cost. The slope of the line is 12, which represents the variable cost.

Conclusion

In this article, we used function equations to represent the total cost to build a dog house. We analyzed the function equation and identified the fixed cost, variable cost, and total cost. We also graphed the function equation and observed that it is a straight line with a positive slope. This mathematical approach provides a clear understanding of the relationship between the cost of the dog house kit, the number of paint cans used, and the total cost.

Discussion Questions

What is the fixed cost of building a dog house?
What is the variable cost of building a dog house?
How does the number of paint cans used affect the total cost?
What is the slope of the graph of the function equation?
What is the y-intercept of the graph of the function equation?

Answer Key

The fixed cost is $\$65$ .
The variable cost is $\$12p$ .
The number of paint cans used affects the total cost linearly.
The slope of the graph is 12.
The y-intercept is 65.

References

[1] "Function Equations" by Math Open Reference. Retrieved from https://www.mathopenref.com/functions.html
[2] "Linear Functions" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f-linear-functions/x2f-linear-functions-article

Additional Resources

[1] "Mathematics for the Nonmathematician" by Morris Kline. Retrieved from https://www.amazon.com/Mathematics-Nonmathematician-Morris-Kline/dp/0486248233
[2] "Calculus for Dummies" by Mark Ryan. Retrieved from https://www.amazon.com/Calculus-Dummies-Mark-Ryan/dp/0470435981
Q&A: Building a Dog House with Function Equations =====================================================

Introduction

In our previous article, we explored the mathematical concept of function equations and applied it to a real-world scenario: building a dog house. We used function equations to represent the total cost of the project and analyzed the relationship between the cost of the dog house kit, the number of paint cans used, and the total cost. In this article, we will answer some frequently asked questions about building a dog house with function equations.

Q: What is the fixed cost of building a dog house?

A: The fixed cost of building a dog house is $\$65$ , which is the cost of the dog house kit.

Q: What is the variable cost of building a dog house?

A: The variable cost of building a dog house is $\$12p$ , where $p$ is the number of paint cans used.

Q: How does the number of paint cans used affect the total cost?

A: The number of paint cans used affects the total cost linearly. For every additional can of paint used, the total cost increases by $\$12$ .

Q: What is the slope of the graph of the function equation?

A: The slope of the graph of the function equation is 12, which represents the rate of change of the total cost with respect to the number of paint cans used.

Q: What is the y-intercept of the graph of the function equation?

A: The y-intercept of the graph of the function equation is 65, which represents the fixed cost of the dog house kit.

Q: Can I use this function equation to calculate the total cost of building a dog house with a different size?

A: Yes, you can use this function equation to calculate the total cost of building a dog house with a different size. However, you will need to adjust the fixed cost and variable cost accordingly.

Q: How can I use this function equation in real-world applications?

A: You can use this function equation in real-world applications such as:

Calculating the total cost of building a dog house with different materials
Determining the optimal number of paint cans to use to minimize the total cost
Comparing the costs of building a dog house with different sizes

Q: What are some limitations of using function equations to calculate the total cost of building a dog house?

A: Some limitations of using function equations to calculate the total cost of building a dog house include:

Assuming a linear relationship between the number of paint cans used and the total cost
Not accounting for other costs such as labor and materials
Not considering the impact of external factors such as inflation and market fluctuations

Conclusion

In this article, we answered some frequently asked questions about building a dog house with function equations. We hope that this article has provided you with a better understanding of the mathematical concepts involved and how they can be applied to real-world scenarios.

Discussion Questions

What are some other real-world applications of function equations?
How can you use function equations to calculate the total cost of building a dog house with different materials?
What are some limitations of using function equations to calculate the total cost of building a dog house?
How can you use function equations to determine the optimal number of paint cans to use to minimize the total cost?
What are some external factors that can impact the total cost of building a dog house?

Answer Key

Some other real-world applications of function equations include calculating the total cost of building a house, determining the optimal number of workers to hire, and predicting the demand for a product.
You can use function equations to calculate the total cost of building a dog house with different materials by adjusting the fixed cost and variable cost accordingly.
Some limitations of using function equations to calculate the total cost of building a dog house include assuming a linear relationship between the number of paint cans used and the total cost, not accounting for other costs such as labor and materials, and not considering the impact of external factors such as inflation and market fluctuations.
You can use function equations to determine the optimal number of paint cans to use to minimize the total cost by finding the minimum value of the function equation.
Some external factors that can impact the total cost of building a dog house include inflation, market fluctuations, and changes in the cost of materials.

References

[1] "Function Equations" by Math Open Reference. Retrieved from https://www.mathopenref.com/functions.html
[2] "Linear Functions" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f-linear-functions/x2f-linear-functions-article
[3] "Mathematics for the Nonmathematician" by Morris Kline. Retrieved from https://www.amazon.com/Mathematics-Nonmathematician-Morris-Kline/dp/0486248233
[4] "Calculus for Dummies" by Mark Ryan. Retrieved from https://www.amazon.com/Calculus-Dummies-Mark-Ryan/dp/0470435981