Solve the problem.
Find the break-even point for the given cost and revenue equations. Round to the nearest whole unit.
[tex]
\begin{array}{l}
C=20 n+134,000 \\
R=160 n
\end{array}
[/tex]
A. 744
B. 180
C. 140
D. 957
Answer (2)
Set the cost equation equal to the revenue equation: 20 n + 134000 = 160 n .
Simplify the equation by subtracting 20 n from both sides: 134000 = 140 n .
Solve for n by dividing both sides by 140: n = 140 134000 = 957.142857... .
Round to the nearest whole unit: 957 .
Explanation
Problem Analysis We are given the cost equation C = 20 n + 134000 and the revenue equation R = 160 n . The break-even point is where the cost equals the revenue, so we need to find the value of n for which C = R .
Setting Cost Equal to Revenue To find the break-even point, we set the cost equation equal to the revenue equation: 20 n + 134000 = 160 n
Isolating the Variable Now, we solve for n . First, subtract 20 n from both sides of the equation: 134000 = 160 n − 20 n
Simplifying the Equation Simplify the equation: 134000 = 140 n
Solving for n Now, divide both sides by 140 to isolate n :
n = 140 134000 n = 957.142857...
Rounding to Nearest Whole Unit Since we need to round to the nearest whole unit, we round 957.142857... to 957 .
Final Answer Therefore, the break-even point is n = 957 units.
Examples
Imagine you are starting a small business selling handmade crafts. The cost equation represents your fixed costs (like rent and equipment) plus the variable cost per item (materials). The revenue equation represents how much money you make per item sold. Finding the break-even point tells you how many items you need to sell to cover all your costs and start making a profit. This concept is crucial for business planning and financial management, helping you determine pricing strategies and production goals to ensure your business is sustainable.
The break-even point is found by setting the cost and revenue equations equal, resulting in approximately 957 units. Thus, the answer is n = 957 units. This means you need to sell 957 units to cover all your costs and begin making profit.
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