- Start with the inequality \(-5b^2 + 450b - 1,000 \geq 9,000\).
- Simplify to \(b^2 - 90b + 2,000 \leq 0\) by subtracting and dividing by \(-5\).
- Use the quadratic formula to find the roots \(40\) and \(50\).
- Determine the valid range is between these roots: \(40 \leq b \leq 50\), which means Sarah must sell between 40 and 50 bouquets.
- The correct statement is: Sarah must sell between 40 and 50 springtime bouquets.
Explanation
- Understanding the Problem
The profit \(P\) from selling \(b\) bouquets is modeled by the function \(P = -5b^2 + 450b - 1,000\). Sarah wants to achieve a profit of at least \(\$9,000\), which can be written as the inequality \(P \geq 9,000\). Our goal is to determine the range of bouquets \(b\) that satisfies this condition.
- Setting up and Simplifying the Inequality
To find the range of \(b\), we start with the inequality:
\[-5b^2 + 450b - 1,000 \geq 9,000\]
Subtracting 9,000 from both sides gives:
\[-5b^2 + 450b - 10,000 \geq 0\]
To simplify, we divide the entire inequality by \(-5\). Remember that dividing or multiplying by a negative number flips the inequality symbol:
\[b^2 - 90b + 2,000 \leq 0\]
- Solving the Quadratic Equation
We now solve the associated quadratic equation \(b^2 - 90b + 2,000 = 0\) to find the roots, which will define the boundaries of our interval. We can use the quadratic formula \(b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}\), where \(A = 1\), \(B = -90\), and \(C = 2,000\):
\[b = \frac{90 \pm \sqrt{(-90)^2 - 4(1)(2,000)}}{2(1)}\]
\[b = \frac{90 \pm \sqrt{8,100 - 8,000}}{2}\]
\[b = \frac{90 \pm \sqrt{100}}{2}\]
\[b = \frac{90 \pm 10}{2}\]
The two solutions are \(b_1 = \frac{90 - 10}{2} = 40\) and \(b_2 = \frac{90 + 10}{2} = 50\).
- Conclusion
Since the parabola \(f(b) = b^2 - 90b + 2,000\) opens upward and we want \(f(b) \leq 0\), the solution is the closed interval between the two roots. Therefore, the range of bouquets Sarah must sell is \(40 \leq b \leq 50\). Comparing this to the given options, the true statement is that Sarah must sell between 40 and 50 springtime bouquets.
Examples
This quadratic model can be used by business owners to determine the optimal production level. By understanding where the profit function crosses certain thresholds, owners like Sarah can decide how many items to manufacture or sell to maximize efficiency and ensure the business meets its financial goals. For example, if the production cost increases, the profit curve shifts, and recalculating these roots helps in adjusting strategies to maintain profitability.