A factory has fixed costs of $1,275 per month. The cost of producing each unit of its product is $2.50. Each unit sells for $10. The equations that model the cost and revenue for producing $x$ units are shown.
Cost: [tex]C (x)=1275+2.5 x[/tex]
Revenue: [tex]R(x)=10 x[/tex]
If the company sells every unit it produces, how many units must it sell in order to break even?
[tex]\square[/tex] units
Answer (2)
The factory must sell 170 units to break even, as determined by setting the cost and revenue functions equal to each other and solving for x . This calculation ensures that total revenue matches total costs. Thus, selling this number of units would mean the factory covers all its expenses without generating profit or loss.
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Set the cost function equal to the revenue function: 1275 + 2.5 x = 10 x .
Simplify the equation: 1275 = 7.5 x .
Solve for x : x = 7.5 1275 .
The company must sell 170 units to break even.
Explanation
Problem Analysis Let's analyze the problem. We are given the cost function C ( x ) = 1275 + 2.5 x and the revenue function R ( x ) = 10 x , where x is the number of units produced and sold. The break-even point occurs when the cost equals the revenue, i.e., C ( x ) = R ( x ) . We need to find the value of x that satisfies this condition.
Setting up the Equation To find the break-even point, we set the cost function equal to the revenue function: 1275 + 2.5 x = 10 x
Isolating x Now, we solve for x . First, subtract 2.5 x from both sides of the equation: 1275 = 10 x − 2.5 x
Simplifying the Equation Simplify the right side: 1275 = 7.5 x
Solving for x Now, divide both sides by 7.5 to isolate x :
x = 7.5 1275
Calculating the Number of Units Calculate the value of x :
x = 170
Final Answer Therefore, the company must sell 170 units to break even.
Examples
Understanding break-even points is crucial for businesses. For example, if you're starting a small business selling handmade jewelry, knowing your fixed costs (like studio rent and equipment) and variable costs (like materials) helps you determine how many pieces you need to sell to cover all your expenses. This calculation guides pricing strategies and production targets, ensuring your business is financially sustainable. By equating costs and revenue, you can find the minimum sales volume needed to avoid losses and start making a profit.
