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.
