A business owner pays $1,200 per month in rent and a total of $120 per hour in employee salary for each hour the store is open. On average, the store brings in $200 in net sales per hour.
Which equations can be solved to determine the break-even point if [tex]C(x)[/tex] represents the cost function, [tex]R(x)[/tex] represents the revenue function, and [tex]x[/tex] the number of hours per month the store is open?
A. [tex]C(x)=1,200+120 x ; R(x)=200 x[/tex]
B. [tex]C(x)=1,200+120 ; R(x)=200 x[/tex]
C. [tex]C(x)=200 x ; R(x)=1,200+120 x[/tex]
D. [tex]C(x)=200 x ; R(x)=1,200+120[/tex]
Answer (2)
The cost function is defined as C(x) = 1200 + 120x, while the revenue function is R(x) = 200x. To find the break-even point where costs equal revenue, we use these equations. Therefore, the correct equations are option A: C(x) = 1200 + 120x ; R(x) = 200x.
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The cost function C(x) includes the fixed monthly rent and the hourly employee salary: C ( x ) = 1200 + 120 x .
The revenue function R(x) is determined by the hourly net sales: R ( x ) = 200 x .
The equations to determine the break-even point are C ( x ) = 1200 + 120 x and R ( x ) = 200 x .
The correct equations are: C ( x ) = 1200 + 120 x ; R ( x ) = 200 x .
Explanation
Analyze the problem Let's analyze the given information to determine the cost and revenue functions. The business owner has a fixed monthly rent of $1 , 200 and pays $120 per hour in employee salary. The store brings in $200 in net sales per hour. We need to find the equations for the cost function C ( x ) and the revenue function R ( x ) , where x is the number of hours the store is open per month.
Determine the cost function The cost function C ( x ) includes the fixed monthly rent and the hourly employee salary. So, the cost function is given by: C ( x ) = 1200 + 120 x
Determine the revenue function The revenue function R ( x ) is determined by the hourly net sales. So, the revenue function is given by: R ( x ) = 200 x
State the equations Therefore, the equations that can be solved to determine the break-even point are: C ( x ) = 1200 + 120 x R ( x ) = 200 x
Examples
Understanding cost and revenue functions is crucial for business owners. For example, a bakery owner can use these functions to determine how many hours they need to keep their shop open each month to cover their expenses (rent, employee salaries) and start making a profit. By setting C ( x ) = R ( x ) , the owner can find the break-even point, which is the number of hours they need to operate to balance costs and revenue. This helps in making informed decisions about staffing, operating hours, and pricing strategies.
