An insurance company claims that for $x$ thousand policies, its monthly revenue in dollars is given by $R(x)=180 x$ and its monthly cost in dollars is given by $C(x)=135 x+12,150$.
a. Find the break-even point.
(Just give me the BREAK-EVEN QUANTITY/point on this one.)
Answer (1)
Set the revenue equal to the cost: 180 x = 135 x + 12150 .
Simplify the equation: 45 x = 12150 .
Solve for x : x = 45 12150 = 270 .
The break-even quantity is 270 .
Explanation
Understanding the Problem We are given the monthly revenue function R ( x ) = 180 x and the monthly cost function C ( x ) = 135 x + 12150 , where x is the number of policies in thousands. The break-even point occurs when the revenue equals the cost, i.e., R ( x ) = C ( 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 revenue function equal to the cost function: R ( x ) = C ( x ) 180 x = 135 x + 12150
Solving for x Now, we solve for x :
180 x − 135 x = 12150 45 x = 12150 x = 45 12150 x = 270
Finding the Break-Even Quantity The break-even quantity is x = 270 . Since x is in thousands of policies, the break-even point is 270,000 policies. However, the question asks for the break-even quantity, which is the value of x .
Examples
Understanding break-even points is crucial for businesses. For example, a small bakery can use this concept to determine how many cakes they need to sell each month to cover their costs, including ingredients, rent, and utilities. By calculating the break-even point, the bakery owner can set realistic sales goals and make informed decisions about pricing and production. This ensures the bakery operates sustainably and avoids losses, contributing to its long-term success.
