Westonci.ca offers quick and accurate answers to your questions. Join our community and get the insights you need today. Experience the ease of finding reliable answers to your questions from a vast community of knowledgeable experts. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
To determine the break-even point for the business, we need to equate the cost function [tex]\( C(x) \)[/tex] and the revenue function [tex]\( R(x) \)[/tex]. Let's derive the costs and revenues step-by-step.
Given:
- Monthly rent is [tex]\(\$ 1,200\)[/tex].
- Employee salary is [tex]\(\$ 120\)[/tex] per hour.
- Revenue or net sales is [tex]\(\$ 200\)[/tex] per hour.
### Cost Function [tex]\( C(x) \)[/tex]
The total cost [tex]\( C(x) \)[/tex] is comprised of the fixed monthly rent and the variable cost of salaries based on the number of hours [tex]\( x \)[/tex] the store is open per month. Therefore, the cost function can be expressed as:
[tex]\[ C(x) = 1,200 + 120x \][/tex]
### Revenue Function [tex]\( R(x) \)[/tex]
The revenue [tex]\( R(x) \)[/tex] is purely based on the number of hours [tex]\( x \)[/tex] the store is open, as the store brings in [tex]\(\$ 200\)[/tex] for each hour. Hence, the revenue function is:
[tex]\[ R(x) = 200x \][/tex]
Now, we have derived the following pairs of functions to determine the break-even point:
1. [tex]\( C(x) = 1,200 + 120x \)[/tex] and [tex]\( R(x) = 200x \)[/tex]
2. [tex]\( C(x) = 1,200 + 120 \)[/tex] and [tex]\( R(x) = 200x \)[/tex]
3. [tex]\( C(x) = 200x \)[/tex] and [tex]\( R(x) = 1,200 + 120x \)[/tex]
4. [tex]\( C(x) = 200x \)[/tex] and [tex]\( R(x) = 1,200 + 120 \)[/tex]
### Determining the Correct Pairs
We will analyze these functions to see which pairs equate the total cost and total revenue accurately:
1. [tex]\( C(x) = 1,200 + 120x \)[/tex] and [tex]\( R(x) = 200x \)[/tex]
This pair represents the correct relationship as:
- The cost [tex]\( C(x) \)[/tex] includes the fixed monthly rent and the variable salary cost.
- The revenue [tex]\( R(x) \)[/tex] depends on the net sales per hour.
2. [tex]\( C(x) = 1,200 + 120 \)[/tex] and [tex]\( R(x) = 200x \)[/tex]
This pair does not make sense from a business standpoint because it represents the cost as simply [tex]\( 1,200 + 120 \)[/tex], ignoring the variable cost component based on hours.
3. [tex]\( C(x) = 200x \)[/tex] and [tex]\( R(x) = 1,200 + 120x \)[/tex]
This pair swaps the roles of cost and revenue, which is not correct, as the monthly rent is not part of the revenue.
4. [tex]\( C(x) = 200x \)[/tex] and [tex]\( R(x) = 1,200 + 120 \)[/tex]
In this pair, the cost [tex]\( C(x) \)[/tex] does not include the fixed monthly rent and the revenue [tex]\( R(x) \)[/tex] is incorrectly represented.
### Concluding the Correct Equations
Based on the analysis, the correct pair of equations to use for determining the break-even point is:
[tex]\[ C(x) = 1,200 + 120x \][/tex]
[tex]\[ R(x) = 200x \][/tex]
These equations accurately represent the cost and revenue functions necessary to solve for the break-even point where the total cost equals the total revenue.
Given:
- Monthly rent is [tex]\(\$ 1,200\)[/tex].
- Employee salary is [tex]\(\$ 120\)[/tex] per hour.
- Revenue or net sales is [tex]\(\$ 200\)[/tex] per hour.
### Cost Function [tex]\( C(x) \)[/tex]
The total cost [tex]\( C(x) \)[/tex] is comprised of the fixed monthly rent and the variable cost of salaries based on the number of hours [tex]\( x \)[/tex] the store is open per month. Therefore, the cost function can be expressed as:
[tex]\[ C(x) = 1,200 + 120x \][/tex]
### Revenue Function [tex]\( R(x) \)[/tex]
The revenue [tex]\( R(x) \)[/tex] is purely based on the number of hours [tex]\( x \)[/tex] the store is open, as the store brings in [tex]\(\$ 200\)[/tex] for each hour. Hence, the revenue function is:
[tex]\[ R(x) = 200x \][/tex]
Now, we have derived the following pairs of functions to determine the break-even point:
1. [tex]\( C(x) = 1,200 + 120x \)[/tex] and [tex]\( R(x) = 200x \)[/tex]
2. [tex]\( C(x) = 1,200 + 120 \)[/tex] and [tex]\( R(x) = 200x \)[/tex]
3. [tex]\( C(x) = 200x \)[/tex] and [tex]\( R(x) = 1,200 + 120x \)[/tex]
4. [tex]\( C(x) = 200x \)[/tex] and [tex]\( R(x) = 1,200 + 120 \)[/tex]
### Determining the Correct Pairs
We will analyze these functions to see which pairs equate the total cost and total revenue accurately:
1. [tex]\( C(x) = 1,200 + 120x \)[/tex] and [tex]\( R(x) = 200x \)[/tex]
This pair represents the correct relationship as:
- The cost [tex]\( C(x) \)[/tex] includes the fixed monthly rent and the variable salary cost.
- The revenue [tex]\( R(x) \)[/tex] depends on the net sales per hour.
2. [tex]\( C(x) = 1,200 + 120 \)[/tex] and [tex]\( R(x) = 200x \)[/tex]
This pair does not make sense from a business standpoint because it represents the cost as simply [tex]\( 1,200 + 120 \)[/tex], ignoring the variable cost component based on hours.
3. [tex]\( C(x) = 200x \)[/tex] and [tex]\( R(x) = 1,200 + 120x \)[/tex]
This pair swaps the roles of cost and revenue, which is not correct, as the monthly rent is not part of the revenue.
4. [tex]\( C(x) = 200x \)[/tex] and [tex]\( R(x) = 1,200 + 120 \)[/tex]
In this pair, the cost [tex]\( C(x) \)[/tex] does not include the fixed monthly rent and the revenue [tex]\( R(x) \)[/tex] is incorrectly represented.
### Concluding the Correct Equations
Based on the analysis, the correct pair of equations to use for determining the break-even point is:
[tex]\[ C(x) = 1,200 + 120x \][/tex]
[tex]\[ R(x) = 200x \][/tex]
These equations accurately represent the cost and revenue functions necessary to solve for the break-even point where the total cost equals the total revenue.
We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
