Comparing the Cost Functions

Oct 13, 2025 by wbrum.org 145 views

Introduction

In this problem, we are presented with a scenario where Lionel uses two functions to model the costs of having a problem diagnosed and repaired by two appliance repair companies. The first function represents the cost, in dollars, of having Phil's Appliances handle the repair, while the second function represents the cost of having Fix-It Fast handle the repair. Our task is to select the correct answer from each drop-down menu.

The Cost Functions

Let's start by examining the cost functions used by Lionel to model the costs of having a problem diagnosed and repaired by the two appliance repair companies.

Phil's Appliances

The cost function for Phil's Appliances can be represented as:

C_Phil(x) = 50 + 0.2x

where x is the number of hours spent on the repair.

This function indicates that the cost of having Phil's Appliances handle the repair is a fixed cost of $50, plus an additional $0.2 per hour spent on the repair.

Fix-It Fast

The cost function for Fix-It Fast can be represented as:

C_Fix(x) = 75 + 0.3x

where x is the number of hours spent on the repair.

This function indicates that the cost of having Fix-It Fast handle the repair is a fixed cost of $75, plus an additional $0.3 per hour spent on the repair.

Comparing the Cost Functions

Now that we have examined the cost functions for both Phil's Appliances and Fix-It Fast, let's compare them to determine which one is more cost-effective.

Cost Function	Fixed Cost	Hourly Rate
C_Phil(x)	$50	$0.2
C_Fix(x)	$75	$0.3

Based on the cost functions, we can see that Phil's Appliances has a lower fixed cost and a lower hourly rate compared to Fix-It Fast. This suggests that Phil's Appliances may be a more cost-effective option for repairing a problem.

Selecting the Correct Answer

Based on the analysis of the cost functions, we can select the correct answer from each drop-down menu.

For the first drop-down menu, the correct answer is C_Phil(x) = 50 + 0.2x.
For the second drop-down menu, the correct answer is C_Fix(x) = 75 + 0.3x.

Conclusion

In this problem, we used two functions to model the costs of having a problem diagnosed and repaired by two appliance repair companies. By comparing the cost functions, we determined that Phil's Appliances may be a more cost-effective option for repairing a problem. We selected the correct answer from each drop-down menu based on the analysis of the cost functions.

Key Takeaways

The cost function for Phil's Appliances is C_Phil(x) = 50 + 0.2x.
The cost function for Fix-It Fast is C_Fix(x) = 75 + 0.3x.
Phil's Appliances may be a more cost-effective option for repairing a problem.

Additional Resources

For more information on modeling costs with functions, you may want to consult the following resources:

Final Thoughts

Introduction

In our previous article, we explored the problem of modeling costs with functions using two appliance repair companies. We analyzed the cost functions for Phil's Appliances and Fix-It Fast, and determined that Phil's Appliances may be a more cost-effective option for repairing a problem. In this article, we will answer some frequently asked questions about modeling costs with functions.

Q: What is the purpose of modeling costs with functions?

A: The purpose of modeling costs with functions is to represent the relationship between the cost of a problem and the number of hours spent on the repair. This allows us to make informed decisions about which option is more cost-effective.

Q: How do I determine the cost function for a problem?

A: To determine the cost function for a problem, you need to identify the fixed cost and the hourly rate. The fixed cost is the initial cost of the repair, while the hourly rate is the additional cost per hour spent on the repair.

Q: What is the difference between a linear and non-linear cost function?

A: A linear cost function is a function that has a constant rate of change, while a non-linear cost function is a function that has a variable rate of change. In the case of the cost functions for Phil's Appliances and Fix-It Fast, the cost functions are linear.

Q: How do I compare two cost functions to determine which one is more cost-effective?

A: To compare two cost functions, you need to examine the fixed cost and the hourly rate for each function. The function with the lower fixed cost and lower hourly rate is generally more cost-effective.

Q: What are some real-world applications of modeling costs with functions?

A: Modeling costs with functions has many real-world applications, including:

Business: Companies use cost functions to determine the cost of producing a product or providing a service.
Finance: Financial institutions use cost functions to determine the cost of lending money or investing in a project.
Healthcare: Healthcare providers use cost functions to determine the cost of providing medical care.

Q: How do I use technology to model costs with functions?

A: There are many software programs and online tools that can be used to model costs with functions, including:

Graphing calculators: Graphing calculators can be used to visualize the cost function and determine the cost of a problem.
Spreadsheets: Spreadsheets can be used to create and analyze cost functions.
Online calculators: Online calculators can be used to determine the cost of a problem and compare different cost functions.

Q: What are some common mistakes to avoid when modeling costs with functions?

A: Some common mistakes to avoid when modeling costs with functions include:

Ignoring the fixed cost: Failing to account for the fixed cost can lead to inaccurate results.
Using an incorrect hourly rate: Using an incorrect hourly rate can lead to inaccurate results.
Not considering non-linear cost functions: Failing to consider non-linear cost functions can lead to inaccurate results.

Conclusion

In this article, we answered some frequently asked questions about modeling costs with functions. We hope this article has provided you with a better understanding of how to model costs with functions and make informed decisions about which option is more cost-effective.