Suppose a gift basket maker incurs costs for a basket according to [tex]$C=9 x+171$[/tex]. If the revenue for the baskets is [tex]$R=28 x$[/tex] where x is the number of baskets made and sold, break even occurs when costs = revenues.
The number of baskets that must be sold to break even is $\square$
Answer (2)
Set the cost function equal to the revenue function: 9 x + 171 = 28 x .
Subtract 9 x from both sides: 171 = 19 x .
Divide both sides by 19: x = 19 171 .
The number of baskets to break even is 9 .
Explanation
Understanding the Problem We are given the cost function C = 9 x + 171 and the revenue function R = 28 x , where x is the number of baskets. We need to find the number of baskets that must be sold to break even, which occurs when the cost equals the revenue, i.e., C = R .
Setting up the Equation To find the break-even point, we set the cost function equal to the revenue function: 9 x + 171 = 28 x
Isolating x Now, we solve for x . First, subtract 9 x from both sides of the equation: 171 = 28 x − 9 x
Simplifying Simplify the right side of the equation: 171 = 19 x
Solving for x Now, divide both sides by 19 to solve for x :
x = 19 171 x = 9
Final Answer Therefore, the number of baskets that must be sold to break even is 9.
Examples
Understanding break-even points is crucial in business. For example, if you're selling lemonade, you need to know how many cups you must sell to cover your costs (lemons, sugar, cups). Similarly, a bakery needs to calculate how many cakes they must sell each day to pay for ingredients, rent, and staff. This concept helps businesses make informed decisions about pricing and production levels to ensure profitability. Knowing the break-even point helps in planning and managing finances effectively.
To break even, the gift basket maker needs to sell 9 baskets, as determined by equating the cost and revenue functions. This calculation involves simple algebra to isolate the variable representing the number of baskets sold. The break-even analysis helps understand the minimum sales required to cover costs.
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