- Identify the break-even condition: \(R(x) = C(x)\), leading to the equation \(x^2 - 45x + 200 = 0\).
- Solve the quadratic equation to find the number of meals sold, \(x = 5\) and \(x = 40\).
- Calculate the corresponding total cost and revenue for these points: \((5, 225)\) and \((40, 400)\).
- Conclude that the break-even points are \((5, 225)\) and \((40, 400)\), which corresponds to option b.
- The final answer is \(\boxed{b}\).
Explanation
- Analysis of the Problem
The break-even point is defined as the level of production where total revenue equals total cost, meaning \(R(x) = C(x)\). We are given the cost function \(C(x) = 5x + 200\) and the revenue function \(R(x) = -x^2 + 50x\). To find the break-even points, we must set these two functions equal to each other:
\[-x^2 + 50x = 5x + 200\]
- Solving for x-coordinates
To find the values of \(x\) at the break-even points, we rearrange the equation \(-x^2 + 50x = 5x + 200\) into a standard quadratic equation of the form \(ax^2 + bx + c = 0\):
\[-x^2 + 45x - 200 = 0\]
Multiplying the entire equation by \(-1\) yields:
\[x^2 - 45x + 200 = 0\]
We can solve this quadratic equation using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[x = \frac{-(-45) \pm \sqrt{(-45)^2 - 4(1)(200)}}{2(1)}\]
\[x = \frac{45 \pm \sqrt{2025 - 800}}{2}\]
\[x = \frac{45 \pm \sqrt{1225}}{2}\]
\[x = \frac{45 \pm 35}{2}\]
This gives us two values for \(x\):
\[x_1 = \frac{45 - 35}{2} = \frac{10}{2} = 5\]
\[x_2 = \frac{45 + 35}{2} = \frac{80}{2} = 40\]
- Calculating Break-even Points and Conclusion
Now that we have the values of \(x\) where break-even occurs, we find the corresponding y-coordinates by substituting these values back into the cost function \(C(x) = 5x + 200\):
For \(x = 5\):
\[C(5) = 5(5) + 200 = 25 + 200 = 225\]
So, the first break-even point is \((5, 225)\).
For \(x = 40\):
\[C(40) = 5(40) + 200 = 200 + 200 = 400\]
So, the second break-even point is \((40, 400)\).
Comparing these points with the given options, we see they correspond to option (b).
Examples
In business, calculating the break-even point is a vital exercise for any entrepreneur. It allows a business owner to determine exactly how many units of a product must be sold to cover all expenses and start making a profit. For instance, if you run a small catering business, knowing the number of meals needed to reach break-even helps you set your pricing and manage your overhead costs effectively. This ensures that you aren't just selling goods, but building a sustainable venture where revenue eventually outpaces costs.