Problem 42 - Algebra 2

Question

BUSINESS A mall owner has determined that the relationship between monthly rent charged for store space \(r\) (in dollars per square foot) and monthly profit \(P(r)\) (in thousands of dollars) can be approximated by the function \(P(r)=-8.1 r^{2}+46.9 r-38.2 .\) Solve each quadratic equation or inequality. Explain what each answer tells about the relationship between monthly rent and profit for this mall.

Step-by-Step Solution

Answer

The rent rates for zero profit are 0.92 and 5.16 dollars per square foot. Max profit is at 2.89 dollars per square foot ($29,480).

Step 1: Understanding the given equation

We start with the quadratic function \( P(r) = -8.1r^2 + 46.9r - 38.2 \) which represents the relationship between the monthly rent \( r \) and profit \( P(r) \). Here, \( P(r) \) is the profit in thousands of dollars, and \( r \) is the rent in dollars per square foot.

Step 2: Finding the vertex of the parabola

Since the given equation is in the form \( ax^2 + bx + c \), where \( a = -8.1 \), \( b = 46.9 \), and \( c = -38.2 \), we find the vertex using the formula \( r = -\frac{b}{2a} \). Substituting gives \( r = -\frac{46.9}{2 \times -8.1} = 2.89 \) dollars per square foot. This value of \( r \) gives the monthly rent rate at which the profit is maximized.

Step 3: Calculating the maximum profit

Substitute the vertex \( r = 2.89 \) back into the original equation to find the maximum profit: \( P(2.89) = -8.1(2.89)^2 + 46.9(2.89) - 38.2 \). Calculating this yields \( P(2.89) = 29.48 \) thousand dollars. So, the maximum profit is approximately $29,480.

Step 4: Solving for break-even points

To find when the profit is zero (break-even points), set \( P(r) = 0 \): \( -8.1r^2 + 46.9r - 38.2 = 0 \). Apply the quadratic formula: \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substituting, we get \( r = \frac{-46.9 \pm \sqrt{46.9^2 - 4 \times -8.1 \times -38.2}}{2 \times -8.1} \). Solving, the roots are approximately \( r_1 = 0.92 \) and \( r_2 = 5.16 \). These are the monthly rent rates where profit is zero.

Key Concepts

Finding the Vertex of a Quadratic Function

Understanding the vertex of a quadratic function is important, especially when it comes to applications like profit maximization. In a quadratic equation of the form \( ax^2 + bx + c\), the vertex represents the peak or lowest point of the parabola, depending on the direction it opens. Here, because the coefficient \(a = -8.1\) is negative, the parabola opens downward, making the vertex the maximum point of the function.

To find the vertex, we use the formula for the \(r\)-coordinate: \(r = -\frac{b}{2a}\). This gives the point at which the rent maximizes the profit in the quadratic context given. As calculated, substituting \(b = 46.9\) and \(a = -8.1\) determines \(r = 2.89\).

This is the optimal rent rate in dollars per square foot that the mall owner should charge to achieve maximum profit. Understanding the vertex helps not only to identify the optimal pricing point but also how changes in prices can affect profits.

Determining Maximum Profit

Upon finding the vertex at \(r = 2.89\), we must now focus on calculating the maximum profit. This is done by substituting this optimal rent back into the quadratic profit function \( P(r) = -8.1r^2 + 46.9r - 38.2 \).

When we place \(2.89\) in for \(r\), the profit function yields \( P(2.89) = 29.48 \) thousand dollars. Therefore, at this rate, the profit reaches its peak, providing approximately $29,480.

Why is this significant? Understanding the concept of maximum profit is crucial for businesses, as it helps strategize rent pricing effectively to ensure they are earning the highest possible profit based on the current market and economic conditions. It's a powerful application of quadratic functions in real-world business scenarios, ensuring that decisions are backed by sound mathematical reasoning.

Identifying Break-even Points

Break-even points are the rent values where the profit becomes zero, meaning no gain or loss at these particular prices. Finding these points provides insight into the minimum and maximum rent rates that the business can tolerate before they start losing money.

To determine these points, solve the equation \( -8.1r^2 + 46.9r - 38.2 = 0 \) using the quadratic formula \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). After substituting the respective values into the formula, the results give the break-even rent prices at approximately \(r_1 = 0.92\) and \(r_2 = 5.16\).

What does this tell us? The business can set rents as low as \(0.92 per square foot and as high as \)5.16 per square foot without incurring a loss. This range is important; it defines the feasible pricing strategy while ensuring profitability remains above zero. Business managers use this to guide their pricing decisions to stay sustainable and competitive. Knowing the break-even points enriches their capacity to mitigate risks associated with pricing strategies.