91Ó°ÊÓ

Profit maximization is all about finding the perfect balance between costs and revenue to achieve the highest possible profit. In this scenario with Lynbrook West, we are dealing with a quadratic profit function: \(P(x) = -10x^2 + 1760x - 50,000\). This function gives us the profit for renting out \(x\) number of apartments. The challenge here is to determine the number of apartments needed to get the largest profit. For a quadratic function that forms a downward-facing parabola, the highest point indicates the maximum profit. To find this point, we need to focus on the vertex of the parabola, which tells us exactly how many apartments Lynbrook West should rent out. Once we have this number, we can then substitute it back into the profit equation to calculate the actual maximum profit value.

The vertex of a parabola is a crucial point that can reveal the maximum or minimum value of a quadratic function. For Lynbrook West's problem, the function \(P(x) = -10x^2 + 1760x - 50,000\) forms a downward parabola because the coefficient \(a\) is negative. This indicates that the vertex holds the maximum point. To find the x-coordinate of the vertex, use the simple formula:
\[ x_v = \frac{-b}{2a} \]
Applying it here with \(a = -10\) and \(b = 1760\):

• Calculate \(x_v = \frac{-1760}{2(-10)} = 88\)

This tells us that renting out 88 apartments will yield the highest profit. Just by knowing the x-coordinate of the vertex, property managers can effectively allocate resources for maximum benefit. Finally, to find the profit at this point, substitute \(x_v = 88\) back into the profit function.

Understanding parabolic equations is essential as they frequently appear in profit functions, making them useful tools in business and economics. A parabolic equation like \(ax^2 + bx + c\) can model cost, revenue, or profit in relation to variable changes—in this case, the number of apartments rented. The general properties of these equations make them particularly helpful:

• A positive \(a\) value results in a parabola that opens upwards, indicating a minimum point.
• A negative \(a\) value, like in Lynbrook West's function, points downwards, revealing a maximum value.

These qualities help us predict outcomes such as maximum profit or optimum resource allocation.Understanding these equations also guides decision-making, such as determining optimal production levels or pricing strategies, providing clarity and direction for achieving the highest benefit.